Tool to provide integrated circuit masks with accurate dimensional compensation of patterns

ABSTRACT

Disclosed are mask definition tools, apparatus, methods, systems and computer program products configured to process data representing a semiconductor fabrication mask. A non-limiting example of a method includes performing a decomposition process on a full Transmission Cross Coefficient (TCC) using coherent optimal coherent systems (OCS) kernels; isolating a residual TCC that remains after some number of coherent kernels are extracted from the full TCC; and performing at least one decomposition process on the residual TCC using at least one loxicoherent system. The loxicoherent system uses a plurality of distinct non-coherent kernel functions and is a compound system containing a paired coherent system and an incoherent system that act in sequence. An output of the coherent system is input as a self-luminous quantity to the incoherent system, and the output of the incoherent system is an output of the loxicoherent system.

CROSS REFERENCE TO RELATED APPLICATION

This patent application is a divisional application of copending U.S.patent application Ser. No. 15/420,246, filed on Jan. 31, 2017, which isa divisional patent application of copending U.S. patent applicationSer. No. 15/329,654 filed Jan. 27, 2017, which is a national stageapplication of International patent application No. PCT/US2016/060462filed Nov. 4, 2016, which claims priority to U.S. provisional patentapplication No. 62/393,866 filed Sep. 13, 2016, and U.S. provisionalpatent application No. 62/259,795 filed Nov. 25, 2015, which are herebyincorporated by reference in their entireties.

TECHNICAL FIELD

The embodiments of this invention relate generally to opticallithography and more specifically relate to optical lithography methodsand systems that use an Optimal Coherent Systems (OCS) approach.

BACKGROUND

The optical micro-lithography process in semiconductor fabrication, alsoknown as the photolithography process, involves the reproduction ofdesired circuit patterns onto semiconductor wafers for an overalldesired circuit performance. The desired circuit patterns are typicallyrepresented as apertures with dimensionally compensated shapes formed ona template commonly referred to as a photomask, where the dimensionalcompensation aims to provide the desired circuit features on the wafer.In optical micro-lithography, patterns on the photo-mask template areprojected onto a photo-resist coated wafer by way of optical imagingthrough an exposure system.

The continuous advancement of VLSI chip manufacturing technology to meetMoore's law of shrinking device dimensions in geometric progression hasspurred the development of Resolution Enhancement Techniques (RET),Optical Proximity Correction (OPC) methodologies, Inverse LithographyTechnology (ILT), and Source Mask Optimization (SMO) in opticalmicrolithography. These techniques aim to provide mask patterns that aredimensionally compensated to correct for the errors that arise whenforming images of mask shapes which are barely resolvable by theprojection optical system. The limited resolution causes the waferlocations where a feature edge is desired in the developed photoresistto actually be exposed by “spillover” light from the images of adjacentfeatures, and the detailed shape of the resulting exposed image must bedetermined in order to provide proper dimensional compensation in themask aperture shapes. The printed wafer shapes are also influenced bynon-ideal development behavior in the photoresist, but this too isdetermined by the detailed image pattern within the neighborhood of agiven feature. The images projected by the optical system are of thepartially coherent kind, meaning that the illumination source patternconsists of many independent illuminating waves rather than a purelycoherent beam, i.e. the source distribution has a complicated shape indirectional space, with the illuminating waves not being so complete intheir directional coverage as to flood-illuminate the mask, which wouldproduce incoherent images. While the directional distribution has acomplex shape which is chosen by methods well known in the art, thetotal intensity field produced by the illuminating waves as they overlapon the mask is generally made highly uniform; thus it is the partialcoherence distribution of the illuminating beams rather than theirintensity distribution which is designed to enhance resolution.

The RET techniques based on partially coherent illumination are expectedto be used by chip manufacturers for the foreseeable future due to thehigh volume yield in manufacturing and extended resolution that theyprovide, and their general past history of success. However, the evershrinking device dimensions combined with the desire to enhance circuitperformance in the deep sub-wavelength domain require ever more computeintensive applications of OPC and related methodologies to ensure thefidelity of mask patterns on the printed wafer as device countsincrease. Methods to provide these capabilities are generally referredto as computational lithography. Device counts in individual integratedcircuit levels now often exceed one billion, and providing dimensionallycompensated patterns on this scale is quite expensive using knownmethods. In recent decades a substantial commercial industry hasdeveloped to implement and apply these methods in an efficient manner.

For the most part all of these methods use the same class of physicalmodel to define the impact of resolution loss in projecting the maskshapes, namely the Hopkins model that is known to govern the complex andnonlinear partially coherent imaging process. In addition, anapproximate but computationally more efficient form of the standardHopkins model is universally applied in order to approximately match thepartially coherent imaging process when mask shapes must be provided atfull chip scale, namely the Optimal Coherent Systems approximation,which, as will be described, approximates the complex partially coherentimaging process by forming a superposition of simpler coherent imagesthat can be determined far more quickly. The underlying physical basisfor these methods is expressed in the well-known Hopkins equation ofpartially coherent imaging, which determines the intensity at a givenimage point from a sum of contributions from all pairs of points in thevicinity of the conjugate mask point, or from all pairs of spatialfrequencies that diffract from within that region of the mask. Becauseof this pairwise interaction, partially coherent imaging takes placewithin a doubled domain (and correspondingly the Hopkins equationoperates over a doubled domain), i.e., partially coherent images are ineffect projected from a doubling of the space in which the mask patternsare defined, and in the Hopkins model the image intensity thus has aquadratic dependence on the pattern content (more specifically abilinear dependence). The bilinear kernel that expresses the imagecontribution from a pair of interfering points or spatial frequencies isknown as the transmission cross coefficient (tcc for interfering pairsof points, or TCC for interfering frequencies). Because of its quadraticnonlinearity the Hopkins equation cannot feasibly be evaluated overpatterns of full chip scale, since compute cost becomes prohibitive.Fortunately, the full chip problem can be reduced to one that scalesnear linearly with area by breaking the circuit level into parts,referred to herein as OPC frames, correction frames, simulation frames,or simply as frames, with these frames being processedquasi-independently using a large number of processors, for example 1024or 2048 processors. These frames must be larger than the resolution ofthe optical system in order that the dimensional compensation providedto a given mask shape properly take into account the influence from allnearby mask shapes that are sufficiently close as to noticeablyinfluence the image of the given shape. (Such influences are referred toas optical proximity effects.) In other contexts the resolution of anoptical system usually refers to, e.g., the width of the central core ofthe lens point spread function, which is about 75 nanometers in modernlithographic systems. However, point spread functions have long tailsthat fall off slowly, and in the context of compensating a mask shapefor optical proximity effects it is therefore necessary to consider theassociated weak impact from relatively distant patterns. The distancerange deemed relevant is referred to in computational lithographyparlance as the ambit, optical ambit, or optical diameter [OD], and istypically 1 or 2 microns. The spatial domain tcc is calculated over thisrange. The size of the OPC frame should, as a minimum necessarycondition, be set at least as large as the optical ambit in order toproperly account for local content when providing dimensionalcompensation. However, sizing the frame at the limiting ambit value isinefficient, because it only provides sufficient buffering context toaccurately compensate features within a very small area at the center ofthe frame. In practice the frame size is therefore set a few timeslarger than the ambit, e.g. the frame size might be set at 5 to 10microns, with the outer region of the frame (within about one OD of theboundary) serving as a guard band. Results from within the guard bandmay simply be discarded, with the inner frame contents being retained,and with the frames being overlapped by e.g. twice the guard band widthin order that each mask pattern falls within the retained region of oneframe.

Since the frames overlap it is not possible to define the contents ofone frame independently of its neighbors, and this enforced overlapallows the shared influence that the inner cores of adjacent frames haveon the wafer image to be accounted for. Though it is generallyimpractical to simultaneously account for the entire network of frameinteractions across a full chip, it is common practice to use multiplecommunicating processors that operate in parallel, so that the task ofdetermining the dimensional compensation for a plurality of frames canbe carried out simultaneously, with the number of interacting framesthat are processed in this way being e.g. 4 or 16. To accomplish thistask the integrated circuit layout may be divided into regions that arecorrespondingly larger than a frame, e.g., 4 or 16 times larger.However, in this approach the basic computational scale of the imagecalculation remains that of the frame rather than the larger region,meaning that the image must be calculated over (typically square) areasof, e.g., 5 microns or 10 microns in size (including guard bands). Suchdimensions are still quite large compared to the core optical resolutionof, e.g., 75 nanometers, and evaluation of the Hopkins equation oversuch areas becomes impractically slow, due to its nonlinear scaling.

To carry out MBOPC (a commonly used acronym for Model-Based OPC) or maskdesign it is therefore necessary to approximately match the images fromthe partially coherent system using simpler systems whose images can becalculated more quickly. In practice all such approaches in current useare variants of the so-called Optimal Coherent Systems (OCS) method,which approximately matches the partially coherent images from thelithographic system with a sum of images produced by predeterminedcoherent systems, to be described in more detail below. The method isvery widely used, but goes by many different names besides OCS, such asthe Optimal Coherent Approximation (OCA), or the Sum of Coherent Systemsmethod (SOCS), or Coherent Decomposition.

In a coherent system the illumination is produced by a single sourcepoint, and so may take the form of a single plane wave once the coherentillumination is collimated onto the mask object. Such illumination byonly a single independent beam causes all pairs of mask points in thedoubled domain to fully interfere with one another, and thus thepairwise interference takes place with a common degree of coherence(namely 100%). Because of this common coherence the image contributionproduced by all interactions of a given mask point with all the othermask points (meaning all points within the other dimension of thedoubled Hopkins domain) can be summed separately, and then multiplied byitself to obtain the image intensity. As a result of this devolution toa single domain, the image amplitude produced by a coherent system (andthus by each coherent system in the OCS approximation of the partiallycoherent imaging system) is formed as a linear superposition ofamplitude contributions from the various mask points (which is thensquared to provide an image intensity), and mathematically this linearsuperposition can be represented as a linear convolution of a coherentkernel with the mask pattern. Linear convolution processes can besimulated very rapidly using Fast Fourier Transforms, meaning thatcalculation of the image contribution from an OCS coherent system can becarried out far more rapidly than direct calculation of the imageproduced by a partially coherent system. Even if the OCS set containshundreds of coherent systems, it can be more efficient whendimensionally compensating mask shapes to approximately match thepartially coherent image by the sum of hundreds of coherent images fromthe OCS set, instead of working directly with the partially coherentintensity. However, the efficiency gain will generally not be largeenough to make dimensional compensation practical at full chip scaleunless a set of coherent systems can be found which successfully matchthe partially coherent images with adequate accuracy using a somewhatsmaller total number of coherent systems, e.g., if the OCS set achievesacceptable accuracy using only about 25 coherent systems.

The simplest approach for choosing coherent systems that match apartially coherent system is to subdivide the partially coherent sourceinto small point-like elements. Point sources provide coherentillumination, and thus a separate coherent system can be defined foreach grid point in a gridding of the complex source shape used by thepartially coherent system. This simple decomposition into coherentsystems was developed by Abbe to analyze microscope images, and is knownas Abbe's method. The illumination for each coherent system essentiallytakes the form of a single plane wave that is incident from thedirection of a particular source point. The projection lens in eachcoherent system collects the coherent light transmitted by the mask, andin the Abbe mode of coherent matching the lens apertures of thesecoherent systems are identical to the lens used by the partiallycoherent system being matched. However, rather than having the differentcoherent systems in the matching set use different illumination tiltsalong with a common lens aperture, one may equivalently use a commondirection for the illuminating plane waves (such as illumination atnormal incidence to the mask), while skewing the lens aperture to adifferent offset position for each of the different coherent systems.These two alternative sets of coherent systems behave equivalentlybecause in Hopkins imaging the effect of tilting a plane wave thatilluminates an object (e.g. the mask) is simply to introduce a matchingdirectional skew or tilt in the plane waves that diffract from the mask,meaning that the set of collected waves can be changed in an equivalentway by either tilting the illumination, or by skewing the collectionaperture to an offset position. Thus, in the Abbe approach each coherentsystem can be formed by shifting the lithographic lens aperture to alocation that is offset to intersect the direction of some single pointin the source, with the intensity contribution from each coherent systembeing weighted by the intensity of the associated source point (and witha common coherent plane wave illumination being used by all systems).Imposition of such a weighting factor is equivalent to introducing auniform change in the transmission of the lens pupil of the coherentsystem.

In some cases this simple Abbe approach can provide an efficiency gainwith the partially coherent sources used in modern lithography, sincecurrent sources are sparse in a relative sense, meaning that currentlithographic sources only introduce significant illuminating intensityfrom a small fraction of the full range of directions from which themask might in principle be illuminated (i.e. only small fraction of thefull hemisphere of potentially incident directions actually containsilluminating waves). In the opposite extreme, i.e. when a mask isflood-illuminated with uniform intensity from a full hemisphere ofdirections, the illumination on the mask object becomes incoherent, andthe pairwise contribution of object points to the image (as specified bythe Hopkins equation) has magnitude zero unless the two points arecoincident, i.e. are the same single mask point. The object effectivelybecomes self-luminous in this incoherent limit, and in this limit thedoubled domain of the Hopkins reduces to a single domain. Incoherentimages can therefore be calculated very rapidly using linear convolutionof an intensity kernel with the self-luminous object. Imaging becomesincoherent when, for example, an object mask is flood-illuminated, orwhen an object is self-luminous, or when a self-luminous pattern iscreated by illuminating a fluorescent medium with a shaped pattern.

Lithographic sources are neither coherent nor incoherent, but they areusually considerably closer to the coherent limit than the incoherentlimit, since the coherence function defined by modern sources showsappreciable content over distances that are distinctly larger than theprojection lens resolution. Nonetheless, most lithographic systemsremain quite far from even the coherent limit. In fact, a significantpractical drawback to the simple Abbe coherent matching approach arisesfrom the relatively large number of coherent systems that are needed tomatch typical partially coherent systems when Abbe decomposition isused. For example, a typical lithographic source shape can easilycontain more than 100 source points that emit with significant intensitywhen an accurate gridding is used, and may contain 100's of additionalpoints that emit with an intensity that is weak but non-zero, whosecontributions should still be included to obtain accurate dimensionalcompensation. Use of such a large number of coherent systems forces anundesirably long compute time when determining appropriate dimensionalcompensations in patterns at full chip scale.

The inefficiency of the simple Abbe form of coherent decompositionarises from the very limited character of the tailoring that this methodmakes when defining each coherent system in the matching set, since theAbbe method attempts to provide a useful contribution to the match bysimply shifting the position and uniform transmission of a lens pupilhaving fixed shape (i.e., in Abbe decomposition each coherent systemaperture maintains the fixed shape of the circular pupil of theprojection lens, except that it is shifted in position and given anadjusted transmission to match the contribution from a single emittingpoint of the partially coherent source).

It is known that a more efficient set of coherent systems can beobtained by employing coherent apertures that have complex general form,wherein the transmission of each pupil is made continuously varying in acomplex pattern that yields the best possible match, rather than beingmerely a simple shifted disk. The transmission pattern of the lensaperture in a coherent system essentially acts as a filter on thediffracted mask spectrum, i.e. the aperture pattern applies a filteringto the mask spatial frequency content that the lens reconverges to thecoherent image, and there is a known method for obtaining the coherentfilter function which best matches the behavior of a partially coherentimaging system.

In particular, a method is known for determining the set of coherentsystem apertures which are optimally efficient, i.e., the set ofaperture transmission functions which will be able to obtain aparticular accuracy level using fewer coherent systems than any otherset of apertures, when averaged over all possible patterns. Sincecoherent kernels are optimal when chosen in this way, they are referredto as Optimal Coherent Systems (OCS), and their use is also referred toas an Optimal Coherent Approximation (OCA), or as a Sum Of CoherentSystems (SOCS) approach. When inverse Fourier transformed to the maskdomain, these optimal coherent pupils become the kernels in linearconvolutions of the mask patterns. It is known that these kernels may beexplicitly determined as the eigenfunctions of the nonlinear(specifically, bilinear) kernel of the Hopkins equation, i.e., aseigenfunctions of the transmission cross coefficient (TCC, in theFourier domain, or tcc, in the spatial domain, with lower case beingused by convention in the latter acronym to denote a spatial domainquantity, and it is further known that the sum of an infinite number ofsquared convolutions of these eigenfunctions with the mask willreproduce the Hopkins equation result exactly. However, in practice OCSmust use only a finite number of such squared convolutions, with eachsquared convolution providing the associated coherent imagecontribution, so that OCS approximately matches the partially coherentimage using a finite sum of coherent images. The eigenfunction kernelsused by OCS may be explicitly determined from the TCC by using standardalgorithms and software packages for eigendecomposition. Some of thesealgorithms provide a complete eigendecomposition of the TCC when the TCCis gridded as a matrix, and since such a decomposition is akin to matrixdiagonalization, the procedure is sometimes referred to as diagonalizingthe TCC. At kernel counts that are practical for OPC (e.g., in the 10 to100 range) the OCS coherent systems (each defined by a single kernel)provide a far more accurate match to the TCC than can coherent systemschosen by the Abbe method.

However, OCS accuracy still entails compromise in practice. For example,it should be noted that in a rigorous treatment the optical interactionrange must be considered unbounded, though the interaction strengthfalls off rapidly to generally negligible levels once the core lensresolution and coherence length are exceeded. If the physical sourcecontains an infinite number of points it would be necessary to use aninfinite number of terms to exactly decompose the TCC into coherentsystem contributions, regardless of whether the Abbe or OCS method isused. However, at practical kernel counts the OCS method exhausts theTCC far more rapidly than the Abbe method (with the former in factexhausting the TCC at the fastest rate possible for coherent systems),and the approximate match to the TCC that OCS provides is thus regardedas a valid decomposition of the exact TCC, even though it generallyleaves unmatched a residual portion of the TCC whose impact on images ofpractical interest is often not entirely negligible.

Despite this imperfect accuracy, the OCS algorithm made MBOPC practical,and broadly speaking it has represented the state of the art in fastsimulation of partially coherent projected images since about themid-1990s. But even though OCS allows computational lithography shapeadjustments to be determined with passable accuracy at speeds that aremany orders of magnitude faster than is possible with direct evaluationof the Hopkins equation, MBOPC at full chip scale still requires verylong compute times (of order one day) on very large computers, and so isquite expensive. Moreover, appreciable accuracy is often sacrificed inorder to mitigate this high computational cost, and this increases theburden on empirical correction procedures that are used to fine-tuneprinted lithographic dimensions during production.

A further difficulty with the OCS algorithm is that the tradeoff betweenaccuracy and speed becomes increasingly less favorable as requiredaccuracy is tightened. Typical industry accuracy requirements haveslowly increased as integrated circuit (IC) feature sizes push closerand closer to fundamental resolution limits, and this improvementrequires a disproportionate increase in the number of OCS systemsemployed.

In summary, conventional practice to control the dimensions of ICpatterns involves adjusting mask patterns in a process whose core is theso-called OCS method. The mask adjustment process (known as opticalproximity correction or OPC) relies on OCS to assess candidate maskadjustments.

During use the OCS method simulates the wafer image of billions of maskfeatures during each iteration of the adjustment. The OCS methodconstructs the wafer image as a sum of coherent images of the mask. Eachcoherent image in this approximate match to the partially coherent imageis obtained as the squared convolution of the mask with a coherentkernel. The kernel may be considered as a function that is used as acomponent of an integration that is repeatedly applied, and each kernelin the OCS set is the inverse Fourier transform of the lens aperture ofan Optimal Coherent System, which may be obtained as an eigenfunction ofthe Hopkins bilinear kernel. In general, a purely coherent image can becalculated as the squared convolution of a kernel with the objecttransmission, with the kernel being the inverse Fourier transform of thelens aperture (e.g. an Airy function in the simple case where the lensaperture is an open circle).

The current OPC practice requires a difficult tradeoff between runtimeand accuracy when employing OCS. The OCS sum is only strictly accurateas an infinite series. However in current practice it may be consideredreasonable to employ about 25 coherent systems to match the partiallycoherent lithographic system with an acceptable balance between runtimeand accuracy, and therefore the OCS sum is typically terminated afteronly about 25 systems. An acceptable compromise can nonetheless involvetoo-large CD errors (typically ˜2 nm although larger errors can beexperienced for some pitches) and too-slow runtimes (e.g., a day or moreon a very large computer).

Clearly, improvements to the conventional OPC and OCS-based methods areneeded.

SUMMARY

In a first non-limiting aspect thereof the embodiments of this inventionprovide a tool that is configured to process input data. The toolcomprises an input to receive input data representing integrated circuitshapes within separate mask regions of a semiconductor fabrication maskfor use in optical lithography; and an output to provide output datarepresenting a mask in which dimensions of mask shapes are compensatedon the basis of image content in the vicinity of each mask shape whenthe mask is projected during optical lithography. The tool is configuredto match a partially coherent lithographic image by superposing a sum ofimages from a set of coherent systems and a sum of images from a set ofloxicoherent systems.

In another non-limiting aspect thereof the embodiments of this inventionprovide a computer-controlled tool that is configured to process inputdata representing integrated circuit patterns of a semiconductorfabrication mask to be used in projection lithography. Thecomputer-controlled tool comprises at least one data processorconfigured to apply a dimensional compensation to circuit pattern shapesbased on an intensity pattern produced in a projected lithographicimage, where the intensity pattern is determined by performing anoptimal coherent systems (OCS) process on input data using coherent OCSkernels derived from at least one Hopkins bilinear Transmission CrossCoefficient (TCC). The at least one data processor is further configuredto perform a decomposition process on data using at least one compoundloxicoherent system in which a constituent coherent system is pairedwith a constituent incoherent system to form the loxicoherent system,and where at least one kernel decomposition is made along an axis thatis slanted between two domains of a Hopkins bilinear model to determinean aperture of the incoherent system.

In a further non-limiting aspect thereof the embodiments of thisinvention provide a tool to process data representing input integratedcircuit patterns of a semiconductor fabrication mask to be used inprojection lithography. The tool comprises a frame generation moduleconfigured to partition each region of a starting mask that is organizedinto separated regions of mask content into overlapped frames of maskdata; a coherent system engine comprised of an optimal coherent systems(OCS) engine having an input to receive the overlapped frames of maskdata of the starting mask and an output to provide a full TransmissionCross Coefficient TCC; and an incoherent system engine having an inputconnected to the output of the OCS engine and an output that provides afinal mask definition for use during fabrication of an integratedcircuit. In the tool a loxicoherent system is comprised of a pair of theOCS engine and the incoherent system engine. The incoherent systemengine is configured to form a residual TCC by removing certain coherentsystem kernels from the full TCC; match the residual TCC with a sum ofmultiplied lower-dimensioned kernels that are separated along axes thatare rotated in a doubled domain between mask content axes in the doubleddomain; decompose at least one low-dimensioned kernel lying within thedoubled-domain in the mean-frequency direction into a product ofcoherent system apertures serving to filter the mask content; select asan intensity kernel at least one low-dimensioned kernel lying along thedoubled domain axis in a difference-frequency direction; and adjust maskfragments by iterating operations across one or more processors.

In a further non-limiting aspect thereof the embodiments of thisinvention provide a method to process data representing a semiconductorfabrication mask. The method comprises performing a decompositionprocess on a full Transmission Cross Coefficient (TCC) using optimalcoherent systems (OCS) kernels; isolating a residual TCC that remainsafter a chosen number of coherent kernels are extracted from the fullTCC; and performing at least one decomposition process on the residualTCC using at least one loxicoherent system.

In a still further non-limiting aspect thereof the embodiments of thisinvention provide a computer-implemented method to process datarepresenting a semiconductor fabrication mask. The computer-implementedmethod comprises performing an Optimal Coherent Systems (OCS) process onthe data using OCS kernels derived from at least one Hopkins bilinearmodel; and performing a decomposition process on the data using at leastone loxicoherent kernel, in which at least one kernel decomposition ismade along an axis that is slanted between two domains of the Hopkinsbilinear model.

In yet another non-limiting aspect thereof the embodiments of thisinvention provide an apparatus that comprises an optimal coherentsystems (OCS) system engine having an input to receive a starting maskand an output to provide a full Transmission Cross Coefficient (TCC).The apparatus further comprises a loxicoherent system engine having aninput connected to the output of the OCS system engine and an output toprovide a mask for use during fabrication of an integrated circuit. Theloxicoherent system engine is configured to form a residual TCC byremoving preferred coherent system kernels from the full TCC; decomposethe residual TCC as a sum of lower-dimensioned kernels that areseparated along axes that are rotated between mask content axes in adoubled domain; decompose at least one low-dimensioned kernel lyingwithin the doubled-domain in the mean-frequency direction into a productof mask filters; select as an intensity kernel at least onelow-dimensioned kernel lying along a doubled-domain axis in adifference-frequency direction; and iteratively adjust mask fragments.

In a further non-limiting aspect thereof the embodiments of thisinvention provide an article of manufacture that comprises a tangiblecomputer readable medium having information stored therein or thereon.The information is configured to convert and transform a first object,embodied as a starting mask, into a second object, embodied as a finalmask which can be used during fabrication of semiconductor circuits andstructures. The information is configured to perform a decompositionprocess on a full Transmission Cross Coefficient (TCC) using optimalcoherent systems (OCS) kernels; isolate a residual TCC that remainsafter a chosen number of coherent kernels are extracted from the fullTCC; and perform at least one decomposition process on the residual TCCusing at least one loxicoherent system.

In another further non-limiting aspect thereof the embodiments of thisinvention provide a data assemblage stored on or in a non-transitorycomputer-readable storage medium. The data assemblage represents maskdata for use in fabricating an integrated circuit, and the dataassemblage is created by a process that comprises performing an OptimalCoherent Systems (OCS) process on the data using OCS kernels derivedfrom at least one Hopkins bilinear model; and performing a decompositionprocess on the data using at least one loxicoherent kernel, in which atleast one kernel decomposition is made along an axis that is slantedbetween two domains of the Hopkins bilinear model.

In one further non-limiting aspect thereof the embodiments of thisinvention provide a tool that is configured to process input data, wherethe tool is comprised of an input to receive input data representingintegrated circuit shapes within separate mask regions of asemiconductor fabrication mask for use in optical lithography; and anoutput to provide output data representing a mask in which dimensions ofmask shapes are compensated on the basis of image content in thevicinity of each mask shape when the mask is projected during opticallithography by a partially coherent imaging system. The tool isconfigured to match a partially coherent lithographic image bysuperposing images from a set of decomposition systems that include aDC-monolinear system.

In a still further non-limiting aspect thereof the embodiments of thisinvention provide a tool configured to process input data andlithographic requirements. The tool comprises an input to receive inputdata representing integrated circuit shapes within separate mask regionsof a semiconductor fabrication mask for use in optical lithography, anda specification of quantitative lithographic goals and requirements as anonlinear programming problem whose variables include mask edgevariables. The tool further comprises an output to provide output datarepresenting a mask in which dimensions of mask shapes are compensatedon the basis of image content in the vicinity of each mask shape whenthe mask is projected during optical lithography by a partially coherentimaging system. The tool is configured to determine image intensitiesfrom values taken by the mask edge variables, with spacings andseparations of the mask edges defining the mask dimensions. The tool isfurther configured to compute the quantitative lithographic goals andrequirements from the image intensities; to adjust the nonlinearprogramming problem variables, including the mask edge variables, todetermine an optimal solution to the nonlinear programming problem; andto determine image intensities produced by the partially coherentimaging system by superposing the images from a set of decompositionsystems that include at least one loxicoherent system.

In yet another non-limiting aspect thereof the embodiments of thisinvention provide a photomask for optical lithography. The photomaskcomprises mask shapes containing a reduced number of edges enabling themask shapes to be decomposed into a reduced number of shots of anelectron beam mask writer that creates the photomask.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

FIG. 1 shows several equations, in FIGS. 1A-1E, that are discussed incontext of the conventional OCS process and the Hopkins optical model.

FIGS. 2A-2C, collectively referred to as FIG. 2, provide backgroundinformation related to this invention, where FIG. 2A explains how thecontributions from each source point to each pair of interfering maskspatial frequencies in the doubled Hopkins domain contribute to imagemodulation, FIG. 2B shows how the sharp pupil of the projection lensdetermines whether both mask frequencies are projected to the wafer, andFIG. 2C illustrates determination of the TCC function that defines theoverall modulation from all source points by means of a Hopkins diagram.

FIG. 3 shows in accordance with the invention that the dependence of theoverall modulation from all source points on the interfering maskfrequencies will exhibit a slope discontinuity as the difference Δfbetween the frequencies passes through zero.

FIG. 4 shows that the slope discontinuity in the TCC occurs along the Δfaxis, whose direction is slanted across the doubled domain of theHopkins equation.

FIG. 5 includes several equations, FIGS. 5A-5C, which show that the TCCis slope-discontinuous in the Δf direction, but is generally continuousin the orthogonal f direction.

FIG. 6 shows an example of the TCC property of slope-discontinuity inthe Δf direction, with continuity obtaining in the orthogonal fdirection, in the case of the TCC that governs imaging with adisk-shaped source.

FIG. 7 includes equations in FIGS. 7A and 7B, which show that a Mercerterm composed of OCS kernels is inherently ill-suited to modeling theslope-discontinuity in the TCC.

FIG. 8 shows an example of the TCC residual used as an element of theinvention, in this case obtained by subtracting a Mercer series from thefull disk-source TCC, showing that the residual after using 11 OCSkernels is strongly concentrated in the area of the slope-discontinuity.

FIGS. 9A and 9B, collectively referred to as FIG. 9, illustrate in FIG.9A a second TCC example for 1D line/space features based on a C-quadsource that is shown in FIG. 9B, where the slope discontinuity in theTCC (also referred to as a “crease”) is clearly visible at Δf=0.

FIGS. 10A and 10B, collectively referred to as FIG. 10, provide acomparison of close-up images of the “crease” in the C-quad TCC whenapproximated using different numbers of OCS systems, namely 24 and 247OCS systems, respectively.

FIG. 11 shows several equations in FIGS. 11A-11C which decompose theresidual TCC in a series of so-called rotated systems in accordance withthe invention, these systems being composed of two differentlower-dimensioned kernels with axes whose direction is rotated betweenthe two domains of the Hopkins equation, also showing that the TCC slopecontributed by each of these systems can, like the TCC itself, be verydifferent in the f and Δf directions.

FIG. 12 shows several equations in FIGS. 12A-12H which are discussed inthe context of finding the two kernels of a rotated system which willbest match a residual TCC.

FIGS. 13A and 13B, collectively referred to as FIG. 13, show example 1Dcross-sections of kernel functions for a rotated system that bestmatches the residual TCC when the full TCC for a C-quad example sourceis approximated using 24 OCS kernels.

FIG. 14 shows the TCC residual that remains after using 24 OCS kernelsto approximate the TCC of the FIG. 9 C-quad example, where the residualTCC error is seen to be strongly concentrated in the area of the slopediscontinuity.

FIGS. 15A-15F, collectively referred to as FIG. 15, show the TCCresidual that remains after 24 OCS kernels are used to approximate theTCC from an example partially coherent lithographic system that uses aso-called free-form or SMO source. Results are shown with and without awafer film stack, and the strong concentration of the TCC residual errornear the slope discontinuity is seen in all cases, with the residualerror exhibiting different characteristic symmetries in its real andimaginary parts.

FIG. 16 includes equations in FIGS. 16A and 16B, which show how onekernel of a rotated system can be determined when the residual TCC erroris strongly concentrated near a slope discontinuity at zero differencefrequency.

FIGS. 17A and 17B, collectively referred to as FIG. 17, shows how thepredominant residual TCC in FIG. 8 from the disk source example can beclosely modeled by the kernels of a rotated system.

FIG. 18 is a simplified block diagram of one exemplary embodiment of adata processing system connected with a mask making machine and alithography and wafer processing system in accordance with theinvention.

FIG. 19 is a diagram of a non-limiting example of an apparatus thatincludes an OCS system engine, and where the OCS system in turn includesa frame generation module and that has an output connected to an inputof a loxicoherent system engine in accordance with embodiments of thisinvention.

FIG. 20 includes equations in FIGS. 20A-20I, which show how a so-calledloxicoherent system element of the invention can be formed by expandinga low-dimensioned kernel of a rotated system into a product of maskfilters that each correspond to the aperture of a constituent coherentsystem, also revising the kernel along the slanted difference frequencydirection to become an intensity kernel of a paired constituentincoherent system, and how such loxicoherent systems can be used torapidly determine the contribution made by a TCC residual to a partiallycoherent image.

FIG. 21 shows a plot of a loxicoherent system response that bestapproximates the TCC residual shown in FIG. 14 for the C-quad sourceexample.

FIG. 22 shows several equations in FIGS. 22A-22J that are discussed inthe context of determining the constituent coherent and incoherentkernels of a first or primary loxicoherent system.

FIG. 23 depicts two plots showing loxicoherent filter kernels calculatedby applying equations of FIGS. 22G and 22J on a discrete grid, withparameter p set to 0.

FIG. 24 shows as an example the reduced residual TCC error obtained bymeans of the invention when a primary loxicoherent system is used toapproximate the residual error which in FIG. 14 was shown to result fromusing 24 coherent systems to match partially coherent images from aC-quad source.

FIG. 25 shows the comparatively large remaining residual TCC error whenthe 24 coherent systems of the C-quad source example are supplemented by2 additional coherent systems instead of the equally compute-intensiveloxicoherent system that was used in FIG. 24 to reduce the residualerror.

FIG. 26 shows an equation which is used in an explanation of why overalllithographic image error is strongly influenced by the accuracy of theoptical model at zero difference frequency, given the typical propertiesof lithographic masks.

FIGS. 27A and 27B, collectively referred to as FIG. 27, plot theintensity present in different spatial frequencies projected by anexample metal level mask, showing the usual situation where most of theenergy is concentrated in the zero order, with much of the remainingenergy being concentrated into directions that project on either the xor y axis of the mask patterns.

FIG. 28 is a Table showing the 1D accuracy of the improvedapparatus/method of this invention versus the conventional approach.

FIG. 29 includes equations in FIGS. 29A-29E, which show that in theasymptotic limit the primary loxicoherent systems of the inventionbecome capable of modeling the entirety of the TCC residual at the slopediscontinuity, whereas each additional coherent system can only capturea small portion of this error, explaining a property that can beexploited in forming higher-order loxicoherent systems.

FIGS. 30A-30C, referred to collectively as FIG. 30, provide a logic flowdiagram illustrating basic steps carried out in accordance with theembodiments of this invention in the non-limiting context of an OPCimplementation. The various blocks shown in FIG. 30 can also be viewedas assemblages of serial-connected and parallel-connectedlogic/arithmetic functional units/modules/engines of at least oneapparatus that implements the tool that this an aspect of thisinvention.

FIGS. 31A and 31B, referred to collectively as FIG. 31, aid inexplaining how patterns in the guard bands that provide mask frames withproper optical context for dimensional compensation are updated betweeniterations to accurately reflect the compensating adjustments made inadjacent frames.

FIG. 32 illustrates plots of two loxicoherent kernels, namely a T″frequency domain filter kernel, and a t′ spatial domain mask filter.

FIG. 33 shows several equations in FIGS. 33A-33C that are discussed inthe context of obtaining the mask filter kernels of a higher-orderloxicoherent system when the residual TCC remaining after extraction ofa first loxicoherent system is concentrated in low but non-zerodifference frequencies.

FIG. 34 shows equations in FIGS. 34A-34M that are discussed in thecontext of obtaining higher-order loxicoherent systems that emphasizeextraction of residual TCC content at quasi-dominant differencefrequencies.

FIG. 35 shows equations in FIGS. 35A-35C that are discussed in thecontext of obtaining higher-order loxicoherent systems that model theimaginary parts of residual TCC errors.

FIGS. 36A-36C, collectively referred to as FIG. 36, show three plots ofreduced levels of imaginary-valued residual TCC error that are obtainedwhen higher-order loxicoherent systems are used according to threedifferent embodiments of the invention to match the imaginary part ofthe residual TCC that is shown as an example in FIG. 15E.

FIG. 37 shows equations in FIGS. 37A-37N that are discussed in thecontext of a homotopy method for obtaining higher-order loxicoherentsystems.

FIG. 38 plots the reduced level of residual TCC error when ahigher-order loxicoherent system obtained in accordance with theinvention by a homotopy method is used to model the example residual TCCerror shown in FIG. 24, with the latter error having already beenreduced by application of a first-order loxicoherent system.

FIGS. 39A and 39B referred to collectively as FIG. 39 illustrategraphically that a slope discontinuity and associated residual error canarise in the TCC at difference frequencies that approach the band limit.

FIG. 40 includes equations in FIGS. 40A-40C which are introduced in adiscussion of the slope discontinuity arising at the band limit, showingthat the resulting residual TCC errors are concentrated at differencefrequencies near twice the numerical aperture in direction cosine units,where they can be addressed by embodiments of the invention.

FIG. 41 includes equations in FIGS. 41A-41G which are presented in adiscussion disclosing a so-called DC-monolinear system of the invention,which efficiently models the portions of the TCC which are mostintensively sampled by typical lithographic masks.

FIGS. 42A-42C referred to collectively as FIG. 42 show example plots ofthe 2D peak or ridge of the predominant residual error in the 4D TCCthat governs the imaging of 2D mask patterns, in the case of a C-quadsource example. These plots illustrate that the regions of worst errorwithin this peak arise from epicycloid-like behavior involving theprojection lens pupil and the source poles.

FIGS. 43A-43C referred to collectively as FIG. 43 show three different2D cross-sections or averaged cross-sections in the 2D manifoldperpendicular to the ridge peak of the predominant residual error in the4D TCC, in the case of a C-quad source example. These plots illustratean azimuthal averaging that occurs when a T″ intensity kernel of aloxicoherent system is determined.

FIG. 44, which includes FIGS. 44A-44D, shows plots of T′ sectors forpredominantly x,y pattern azimuths and T′ sectors for predominantly 45°orientations, where the sectoring is implemented to counteract anazimuthal averaging in T″ kernels for a full 4D error residual.

FIG. 45 shows an equation, which is used in the context of defining T″intensity kernels of the invention which, in conjunction with sectoredT′ kernels, counteract azimuthal averaging in modeling 4D errorresiduals, also taking advantage of the spectral inhomogeneity oftypical IC mask patterns.

FIG. 46 shows equations in FIGS. 46A-46D, which illustrate that when aplurality of loxicoherent systems are used, the constituent intensitykernels of these loxicoherent systems can be simultaneously optimized toreflect their joint use, showing explicitly how this is done in the casewhere two loxicoherent systems are employed.

FIG. 47 includes equations in FIGS. 47A-47L which are introduced in adiscussion of how the novel decomposition systems of the invention maybe used during inverse lithography procedures to more efficientlycalculate cost functions or augmented Lagrangians, and their gradients.

DETAILED DESCRIPTION

The word “exemplary” is used herein to mean “serving as an example,instance, or illustration.” Any embodiment described herein as“exemplary” is not necessarily to be construed as preferred oradvantageous over other embodiments. All of the embodiments described inthis Detailed Description are exemplary embodiments provided to enablepersons skilled in the art to make or use the invention and not to limitthe scope of the invention.

As employed herein a “loxicoherent system” is a term/phrase derived tobe descriptive of embodiments of this invention. “Loxi” is Greek forslanted, the significance of which will be explained in detail belowwith respect to a loxicoherent kernel. The structure of a loxicoherentsystem is novel in that each such system is a compound system comprisedof a plurality of constituent systems, for example a pair of constituentsystems operating in sequence. The sequence begins with a firstconstituent system of the pair that is a coherent system, and thatimages the mask amplitude under plane wave illumination through apredetermined lens aperture (represented computationally by apredetermined filtering kernel) to produce a coherent image whosesquared amplitude constitutes an intensity. This intensity is thenpropagated through a constituent incoherent system of the pair ofsystems, this constituent incoherent system having its own predeterminedkernel (in particular an intensity kernel), thus forming an output imagethat serves as the output image of the loxicoherent system as a whole.The output image contribution from at least one loxicoherent system issummed with the contributions from other employed systems toapproximately match the partially coherent lithographic image. Themathematical structure of the loxicoherent system is novel and clearlydifferent from that of prior art coherent systems at least for thereason that it contains two distinct kernel functions rather than one(although below there are references to the T′ function that sometimesterm it to be a filter or aperture function rather than a kernel),corresponding to the two distinct apertures of the paired coherent andincoherent systems that comprise the loxicoherent system. It is notedthat, while T′ certainly qualifies as a kernel even though sometimes itis referred to as a filter or aperture, it can be described in any ofthese ways.

Aspects of the embodiments of this invention pertain to thedecomposition of the TCC using non-coherent kernels for fastercalculation of lithographic images. Conventional OPC codes achievepractical compute times at full-chip scale by approximating partiallycoherent images as sums of coherent images, a methodology knownvariously as OCA, SOCS, or OCS. Though many refinements have been madeto the OCA methodology since its introduction in the mid-1990s, thebasic approach of decomposing the partially coherent Hopkins kernel(TCC) as a sum of coherent systems has remained the state of the art fortwo decades.

An aspect of this invention is to derive and demonstrate a new form ofimage decomposition that is designed to closely match those portions ofthe TCC which are most recalcitrant to standard OCA. The newdecomposition systems can be referred to herein as being loxicoherent.

While coherent systems employ a single convolution kernel, eachloxicoherent system uses at least two distinct kernels. As with standardcoherent systems, compute time with loxicoherent systems is proportional(with some overhead) to the number of kernel convolutions. Tests withone dimensional (1D) patterns show that for a given kernel-count budgetin the typical, e.g., 10-100 range, image calculation error canroutinely be reduced by at least a factor of five if loxicoherentsystems are used in the decomposition. Loxicoherent systems likewiseenable a given worst-case accuracy target to be achieved with at leastthree times fewer kernels. Based on theoretical arguments one may expectfor 2D systems that the speed/accuracy tradeoff will remain far superiorto that of standard OCA, although possibly by a smaller margin ascompared with 1D patterns.

Standard OCA kernels correspond to the pupils of coherent imagingsystems. The output of a coherent system is linear in amplitude, whereasa loxicoherent system has a more complex structure that is entirelynonlinear even in the lowest-order term. The structure of loxicoherentsystems will be explained in detail herein, and they will be shown to bewell-suited for extraction of any near-Toeplitz components present inthe TCC. Such components have an eigenvalue spectrum that decays veryslowly, and so are difficult to capture with OCA.

It can be shown that TCCs for lithographic systems in fact containstrong Toeplitz-like components that arise from slope discontinuitiesassociated with the sharp aperture of the projection lens.Asymptotically, the uncaptured TCC becomes dominated by suchdiscontinuities, and under idealized assumptions the fractional portionof the remaining un-mapped TCC that each new OCA kernel is able toextract becomes arbitrarily small, in the limit where a very largenumber of kernels has already been extracted. In contrast, a singleloxicoherent system is able to capture the entire remainder in thisidealized limit. While these behaviors apply in an asymptotic regimethat can never be fully realized, qualitatively similar behavior is seenwith practical kernel counts.

The rich structure of loxicoherent systems makes them useful formatching recalcitrant portions of the TCC, but their increasedcomplexity may also make them more difficult to determine optimally inthe general case. However, the largest practical benefit arises in thespecial case where the loxicoherent system must fit the TCC remainderleft uncaptured by a typical set of OCA coherent kernels. In thisspecial case a fast analytic method for choosing optimal loxicoherentkernels compares very favorably to brute-force numerical optimization.

A loxicoherent system kernel which is least-squares optimal can berigorously obtained under general conditions, analogous to choosing aTCC eigenfunction as the least-squares optimal (lone) kernel of acoherent system. However, this rigorous method only optimizes a singlekernel in the loxicoherent system, and optimization of all constituentkernels is necessary to obtain full advantage from the loxicoherentstructure. In many cases of practical importance it proves possible todetermine all constituent kernels by combining quasi-analyticcalculations with fast (linear) least-squares fits. Under generalconditions, illustrated herein for an exemplary 1D embodiment, ahomotopy algorithm has been found to reliably produce an accurate andcomplete set of kernels.

Loxicoherent systems can improve accuracy during Inverse LithographyTechnology (ILT) as well as OPC. By using adjoint differentiation thegradient of a cost function or Augmented Lagrangian can be calculatedwith the same FFT-gated near-linear area scaling that the forwardintensity calculation exhibits.

A non-limiting aspect of this invention is a mask design andconfiguration tool that provides a mask having dimensionally compensatedshapes. The tool is configured to process data from an input datastream, input database, or input queue, where the data representsintegrated circuit shapes within separate mask regions of asemiconductor fabrication mask for use in optical lithography. The toolproduces an output database or output data stream in which thedimensions of the mask shapes are compensated on the basis of the imagecontent in the vicinity of each shape when the mask is projected duringoptical lithography. In operation the tool matches a partially coherentlithographic image by superposing a sum of images from a set of coherentsystems and a sum of images from a set of loxicoherent systems, althoughthe image from a single loxicoherent system may be used instead of a sumof contributions from a plurality of loxicoherent systems. A primaryinput for both sets of systems is a frame of integrated circuit shapesconstituting a portion of a mask region from the input queue. Inpreferred embodiments each loxicoherent system is a compound systemcomprising a paired coherent system and incoherent system that act insequence, with the output of the constituent coherent system being inputas a self-luminous quantity to the constituent incoherent system, andwith the output of the incoherent system then serving as the output ofthe loxicoherent system.

The lens apertures in the coherent system set may be the Fouriertransforms of optimal coherent systems (OCS) kernels obtained bycarrying out an eigendecomposition process on a full transmission crosscoefficient (tcc, or TCC in the frequency domain). The apertures of theconstituent coherent and incoherent systems in each compound system ofthe loxicoherent system set may be obtained by isolating a residual TCCthat remains after the chosen set of coherent kernels in the coherentsystem set are extracted from the full TCC; and then performing at leastone decomposition process on the residual TCC using at least oneloxicoherent system. In preferred embodiments each loxicoherent systemis chosen to closely match the portion of the TCC that remains unmatchedby all previously chosen systems, and in preferred embodiments the firstloxicoherent system (also referred to as the primary loxicoherentsystem) matches portions of the TCC that are recalcitrant to matching byOCS systems. In general, a system or set of systems that makes anapproximate match is considered to extract the portion of the TCC thatit matches, with the remaining portion of the TCC forming a residualTCC. The matched TCC portion is itself a TCC that can be used in theHopkins equation to determine the total intensity produced by the systemor set of systems that perform the approximate match. Such a TCC can bereferred to as an approximate TCC.

In another aspect thereof the exemplary embodiments of this inventionprovide a computer-controlled tool to process an input stream ofintegrated circuit patterns representing a semiconductor fabricationmask to be used in projection lithography. The tool applies adimensional compensation to the shapes based on the intensity patternproduced in the projected lithographic image, with this intensity beingdetermined by first carrying out an Optimal Coherent Systems (OCS)process on the data using coherent OCS kernels derived from at least oneHopkins bilinear TCC; and then performing an additional decompositionprocess on the data using at least one system of a type referred toherein as loxicoherent, in which a constituent coherent system is pairedwith a constituent incoherent system to form the loxicoherent system,and where at least one kernel decomposition is made along an axis thatis slanted between two domains of the Hopkins bilinear model todetermine the aperture of the incoherent system.

It should be noted that the shape distortion which arises in waferimages of projected masks will in general entail a positional shift in,e.g., the center of gravity of a printed feature (relative to thenominal image conjugate), as well as changes in e.g. the width or lengthof the printed feature. Dimensional compensation of the mask featuresshould preferably include corrective positioning of the printed featureas well as corrective sizing, i.e., dimensional compensation involvesthe proper positioning of the edges of each printed feature, as well asthe achievement of proper feature sizes. Thus, dimensional compensationmay be understood as requiring the proper spacing between the printedfeatures, as well as requiring the proper dimensions within eachfeature. Similarly, the mask dimensions which receive compensationinclude the spacings between features as well as the widths of features(i.e. all polarities of edge separation are included), and dimensionalcompensation may equivalently be understood as a set of compensatingadjustments that are made in the positions of the edges of maskfeatures.

In another aspect thereof the embodiments of this invention provide atool to process data representing input integrated circuit patterns of asemiconductor fabrication mask to be used in projection lithography. Thetool comprises an OCS system engine having an input to receive astarting mask organized into separated regions of mask content and anoutput to provide a full TCC. The tool includes a module/function thatdivides each region into overlapped frames of mask data. The toolfurther includes a loxicoherent system engine having an input connectedto the output of the OCS system engine and an output that provides afinal mask definition for use during fabrication of an integratedcircuit. The loxicoherent system engine is configured to form a residualTCC by removing certain coherent system kernels from the full TCC; tomatch the residual TCC with a sum of multiplied lower-dimensionedkernels that are separated along axes that are rotated in a doubleddomain between mask content axes in the doubled domain; to decompose atleast one low-dimensioned kernel lying within the doubled-domain in themean-frequency direction into a product of coherent system aperturesserving to filter the mask content; to select as an intensity kernel atleast one low-dimensioned kernel lying along the doubled domain axis ina difference-frequency direction; and to adjust mask fragments byiterating operations across one or more processors. The iteratedoperations include determining loxicoherent system contributions to animage intensity at target edge positions by applying incoherentintensity kernels to squared mask transmissions through the coherentsystem that have been filtered by the mask filters; by determining theimage intensity at target edge positions by adding the loxicoherentcontributions to the sum of intensities from the coherent systems; bymoving mask fragments adjacent to target edge positions whose intensityis lower than the intensity at the edge of an anchoring feature in adirection towards a darker side of the adjacent target edge; by movingmask fragments adjacent to target edge positions whose intensity ishigher than the intensity at the edge of the anchoring feature in adirection towards a brighter side of the adjacent target edge; bymodifying edge positions within frame overlap regions to reconcile theposition movements made in the frames that overlap; and by terminatingthe mask fragment adjustment when the intensities at all target edgepositions match that of the anchoring feature to within a tolerancevalue.

In the embodiments of this invention a decomposition process isperformed on a full transmission cross coefficient (TCC) using optimalcoherent system (OCS) kernels. The process involves isolating a residualTCC that remains after a chosen number of coherent systems are extractedfrom the full TCC; and performing at least one decomposition process onthe residual TCC using at least one incoherent system that operates withan intensity kernel.

It should be understood that in some embodiments of this invention thetool can be embodied in whole or in part as a method or as an apparatus,or as a combination of a method and an apparatus.

It should also be understood that in some embodiments of this inventionthe tool can be embodied, in whole or in part, as an article ofmanufacture that comprises a tangible, non-transitory computer readablemedium having information stored therein or thereon. The information isconfigured to convert and transform a first object, embodied as astarting mask, into a second object, embodied as a final mask that canbe used during fabrication of semiconductor circuits and structures. Theinformation is configured to perform a decomposition process on a fulltransmission cross coefficient (TCC) using optimal coherent system (OCS)kernels; isolate a residual TCC that remains after a chosen number ofcoherent systems are extracted from the full TCC; and perform at leastone decomposition process on the residual TCC using at least oneincoherent system that operates with an intensity kernel.

It should also be understood that in some embodiments of this inventionthe tool can be embodied, in whole or in part, as a data assemblage thatrepresents mask data for use in fabricating an integrated circuit, wherethe data assemblage can be stored on or in a computer-readable datastorage medium. The data assemblage is created by a process thatcomprises performing an Optimal Coherent Systems (OCS) process on datausing OCS kernels derived from at least one Hopkins bilinear TCC; andperforming a decomposition process on the data using at least oneloxicoherent kernel, in which at least one kernel decomposition is madealong an axis that is slanted between two domains of the Hopkinsbilinear TCC.

Before describing the embodiments of this invention in further detail,and by way of introduction, in recent decades OPC has become a criticalstep in integrated circuit (IC) manufacture. The use of OPC only becamefeasible because the so-called OCS method allows partially coherentimages to be calculated over large areas in near-linear time. However,OCS is an approximation whose residual errors under practical cutoffscan amount to a few nanometers.

A key accuracy limitation of OCS arises from limiting the number ofcoherent systems in the matching set, which is a key step in making OCScomputationally feasible. Since each coherent system uses a singlepredetermined aperture that is represented computationally by a singlekernel (most commonly an eigenfunction of the TCC), it follows thatlimiting the number of coherent systems for the sake of efficiency iscomputationally equivalent to truncating OCS kernel count, for example,limiting the number of TCC eigenfunctions employed. The embodiments ofthis invention are directed in part towards addressing the error thatarises from truncating the kernel count.

The embodiments of this invention use novel loxicoherent systems whosekernels are generated by decomposing a transmission cross coefficient(TCC) of a lithographic system in new ways, namely into systems whicheach include a plurality of distinct kernels, in contrast with prior artOCS systems that are each formed with a single type of kernel, namelythe single kernel that describes the transmission of a coherent systemaperture. In exemplary embodiments the new loxicoherent systems includea paired coherent system and incoherent system acting in sequence, eachrepresented by its own distinct kernel. In exemplary embodiments the newloxicoherent kernels are separated along axes which are rotated into a(non-spatial) direction that is skewed (i.e. slanted) between the duo ofconventional mask manifolds whose coordinates are paired to form the 4DHopkins domain, as will be discussed.

It will be understood that “rotation” does not refer here to aconventional geometrical rotation between the x and y axes of thecircuit shapes, but rather this rotation takes place in the moreabstract higher-dimensioned Hopkins domain that is formed as a doublingof the xy plane of the mask content.

A loxicoherent kernel that has been separated along an axis lying in adirection that is skewed between these two mask content planes is highlyefficient at extracting the TCC content that is recalcitrant torendition using standard OCS kernels. (As was noted above, “Loxi” isGreek for slanted.) It should also be understood that because of thehigher dimensionality involved, the “axis” on which the loxicoherentkernel is separated may actually be two-dimensional, i.e. containing xand y components that correspond to the x and y axes of the circuitshapes. The separation direction of these kernels will sometimes bereferred to as “diagonal”, descriptive in the same way as “rotated” or“slanted”, but here again the term “diagonal” should not be interpretedin an overly literal way.

The prior art OCS method matches the partially coherent lithographicimage using a sum of images produced by optical systems that are purelycoherent. In contrast, the optical systems used by the invention tomatch partially coherent images include at least one compound systemcomprising a plurality of constituent systems, such as paired coherentand incoherent constituent systems that operate in sequence to produce,as a final output from the pair, a non-coherent contribution to theimage match. The images which are summed during application of OCS areconventionally referred to as coherent system images, and similarly thenovel summed images formed in accordance with the invention by eachpaired coherent and incoherent system will be referred to asloxicoherent system images, and further each such paired coherent andincoherent system will be referred to as a loxicoherent system. Whilecoherent systems are linear in the input amplitude and incoherentsystems are linear in the input intensity, the compound structure ofloxicoherent systems makes them fully nonlinear; however, the linearityof their constituent systems allows the output from loxicoherent systemsto be determined with relatively low compute cost. Computationally, thecoherent images used in OCS are determined by computing Mercer seriesterms that are composed from coherent kernels, with these mathematicalseries terms themselves being referred to in the art as “coherentsystems”. In analogy with this convention, the mathematical structuresformed in accordance with the invention to calculate images fromloxicoherent kernels will similarly be themselves referred to as“loxicoherent systems”. The specialized meaning of the term “system” inthis context will be clear to those skilled in the art; in the contextof image decomposition the term “system” may be taken to refer generallyto a term in a decomposition series where the term itself represents anoptical system computationally, with this decomposition term beingformed from a kernel or a plurality of kernels. “System” may, of course,also refer to the optical system whose behavior is describedcomputationally by the decomposition series term (with the kernelfunctions being related to the lens aperture transmission of the opticalsystem by well-known physical laws). As another point of nomenclature, afinite set of matching systems that only achieve an approximate matchingis nonetheless considered to provide a decomposition of the partiallycoherent image produced by a given mask, and, more generally, such a setof decomposition systems is still described as providing a decompositionof the partially coherent TCC, even though such a system set leaves aresidual TCC unaccounted for that may have non-negligible magnitude.

In the prior art OCS decomposition each coherent system term is formedfrom two copies of the same frequency-domain kernel function, since theterm represents the physical behavior of a coherent imaging system whoseaperture has the amplitude transmission pattern specified by the kernel,and the squaring of the image amplitude to provide the intensity causesthis same single kernel to be repeated along each of the orthogonal maskaxes of the Hopkins domain. In contrast, each decomposition system termof the invention is formed from at least two distinct kernel functions.

The invention employs a set of coherent systems, as well as one or moreloxicoherent system(s). DC-monolinear systems, discussed in detailbelow, are also compound, and also contain two distinct kernelfunctions. One can consider a DC-monolinear system to be a specific typeof loxicoherent system since it meets these criteria. If one adopts thatconvention one could simply use the phrase “loxicoherent decompositionsystem term”—it being noted that the constituent kernels of theDC-monolinear system do not meet the narrower criteria of operatingsequentially or of lying along rotated axes.

In some non-limiting embodiments these new systems use three kernelsinstead of two, and some of these kernels are functions that lie alongrotated or non-orthogonal axes within the Hopkins domain. In somenon-limiting embodiments the loxicoherent system uses two copies of afirst distinct kernel to represent a coherent system that is pairedsequentially with an incoherent system which uses a single copy of asecond distinct kernel, so that the loxicoherent system as a wholeemploys two distinct kernels. One general aspect of this invention thatdistinguishes it from prior systems is that it uses systems that areformed from more than one kind of kernel.

Computationally, loxicoherent systems have a richer structure than thebilinear product form (Mercer form) of OCS kernels, for examplecombining multiple distinct kernels in a triple product system, but likeOCS kernels they can be applied using convolutions that exploit theefficient scaling of Fast Fourier Transforms (FFTs). The embodiments ofthis invention address and mitigate the limitations of standardlithographic practice related to the stringent tradeoff between accuracyand computational cost that is faced whenever the OCS method is used inimage calculations, particularly the image calculations that must bemade in order to produce functional lithographic masks. The accuracygain from employment of increased numbers of traditional OCS systems isfound to face diminishing returns, due to content in the exact TCC thathas slope-discontinuities which are a consequence of the sharp bandlimit cutoff of lithographic lenses, the slope-discontinuities makingthis content extremely recalcitrant to matching with standard OCSkernels. The loxicoherent systems of the invention can be tuned toefficiently extract this content, along with other portions of the TCCthat a standard OCS expansion does not capture with a frugal number ofterms. The novel decomposition systems of the invention can also betuned to accurately extract TCC regions that are heavily sampled bycritical or predominant mask content, such as regions where one of thetwo interfering frequencies in the doubled domain is DC, these regionsbeing important because the DC order is usually very strong inlithographic masks. The adoption of loxicoherent systems allows a givenaccuracy target to be achieved more efficiently than is possible byincreasing the number of conventional OCS systems, and the accuracytarget can be realized with fewer total systems being used. In otherwords, the cost-accuracy tradeoff inherent to OCS image simulation issignificantly mitigated by the use of this invention.

Reference is made to FIG. 1 to provide some background information,including prior art computational considerations, related to thisinvention.

Lithographic imaging employs partially coherent illumination, whereinsource points that are deployed in a complex tailored directionalpattern illuminate a mask, with each source point illuminating the maskwith a coherent plane wave that provides a unique slew in phase acrossthe different mask points due to the specific directional tilt of thewavefront emitted by the source point. As a result of the ensemble ofdifferent illuminating tilt phases from the totality of chosen sourcedirections, the mask points interfere with one another to a varyingdegree depending on their relative position, causing the opticallyinterfering mask content from each pair of mask points to make its own(generally) unique contribution to the image. This standard form ofpartially coherent imaging can be considered to be governed by theHopkins equation (see FIG. 1A in FIG. 1). The Hopkins equation isbilinear, i.e., it involves a quadratic double convolution over the maskpattern m(x), using a kernel referred to here as the “tcc”, which is aspatial domain version of the Transmission Cross Coefficient, also knownas the Hopkins C function. The tcc may also be regarded as an operator.The Hopkins equation shows that the intensity at each point in the imageis given by a sum (or integration) of contributions from all differentpairs of points on the exposing mask, rather than being a simple sum ofcontributions from all mask points; hence the Hopkins equation involvesa doubled domain, and is not a linear convolution. For simplicity themask transmission m(x) (or reflectivity, in the case of a reflectingmask) is written in FIG. 1A (and in many other equations to follow) as afunction of a single position coordinate x; however it is understoodthat the mask transmission is in general a function of both x and y,although the most critical features for IC performance are often 1Dgrating-like or line-like patterns. As a kernel that is integrated overpairs of points, the tcc has as its domain a doubly-dimensioned spacethat expands the dimensionality of the mask transmission m(x) twofold,e.g. to 4D for 2D mask patterns. This space will be referred to as theHopkins domain or doubled domain, as will the frequency space of theFourier transform of the tcc. The tcc is, in fact, most commonlyexpressed as a Fourier transformed quantity, which we will denote as theTCC; more specifically, the quantity TCC(f₁,f₂) is obtained by Fouriertransforming tcc(x₁,x₂) in both of its arguments. In the convention usedhere for operator kernels like the tcc, the exponential Fourier factorfor the 2nd argument is inverted in sign. In capitalizing the Fouriertransform of the spatial-domain tcc (i.e., the TCC) we also follow aconvention in which lower case denotes a spatial domain version of aquantity and upper case the frequency domain version, e.g., for the maskm(x) versus M(f). Each individual mask frequency can for simplicity beconsidered to propagate as a plane-wave whenever the mask is illuminatedby a single source point. In 1D each pair of interfering frequenciesproduces a sinewave intensity modulation trace in the image (with asuperposed overall phase skew in general). Similarly, in 2D theinterfering pair produces extended sinewave fringes that may have anarbitrary orientation in the x,y plane. The Hopkins equation shows thateven when the mask pattern contains a great many spatial frequencies, itis only the action of the optical system on pairs of frequencies thataffects the image. However, even the two-fold expansion of the domain ofinteraction that is present with partially coherent imaging iscomputationally very expensive to evaluate in the physically accuratemanner specified by the Hopkins equation.

Fortunately lithography practitioners have made use of the OCS method toapproximate partially coherent imaging; OCS can reduce the computationalburden by roughly ˜10³ under typical conditions. As is shown in theequation of FIG. 1B in FIG. 1 the OCS method expands tcc(x₁,x₂) in aMercer series that is strictly accurate as an infinite series, but thatmust be truncated after only a relatively small number of terms in orderto hold OPC runtimes to an acceptable duration. Under typical currentpractice, the equation of FIG. 1B might be truncated after ˜25 terms.Each term of FIG. 1B, denoted t_(OCS), is responsible for the imagecontribution produced by one coherent system in the approximatelymatching set. Each coherent system essentially approximates the equationof FIG. 1A as a product of separate linear convolutions (actually asquare), as shown in the equation of FIG. 1C. Each term in FIG. 1C hasthe form of a squared convolution of a kernel with the masktransmission.

The equation of FIG. 1A shows that the partially coherent image of aspecified mask m(x) is entirely determined by the tcc functionassociated with the lithographic optical configuration being used. Oneconsequence is that any set of decomposition systems which canaccurately approximate the tcc will also be able to accurately match theimages produced by the lithographic optical system. The converse neednot be true in situations where the mask patterns m(x) are highlyrestricted in form, since the images from a limited set of patterns maynot be strongly impacted by all portions of the tcc. However, in manycases of practical interest the mask patterns contain considerablediversity, particularly when the set of e.g. 10¹⁰ patterns comprising afull integrated circuit level is considered, and failure to accuratelycompensate the dimensions of even a single pattern during OPC can causethe entire chip to fail. Moreover, this diversity of potential inputsincreases if a given decomposition system is applied to multipledifferent masks. Most embodiments of the invention are designed toaccurately match the tcc of a specified partially coherent imagingsystem (or equivalently, the TCC), though, as will be discussed, someembodiments tune the matching to emphasize accuracy for objects thatembody particular traits of lithographic masks, and some embodimentsallow tuning to match lithographic mask content of particular strategicimportance. Prior art coherent system decomposition also aims toapproximate the tcc as a whole, in most cases.

The coherent system kernels used in the FIG. 1C convolutions are theso-called OCS kernels ψ_(j)(x). Ordinarily these kernels are calculatedas the eigenfunctions of the tcc kernel, or as the inverse Fouriertransforms of the eigenfunctions of the TCC kernel. A commonmathematical convention for eigendecomposition requires that the norm ofthe eigenfunctions be set to 1. If that convention is followed, themathematical expression for a Mercer series representation of anoperator will include the eigenvalues as well as the eigenfunctions ofthe operator. A convention that can be more convenient in lithographyapplications is to normalize the eigenfunctions of the tcc operator tohave a norm equal to the square root of the associated eigenvalue, andthis latter convention will be used in describing the computations madeby the invention, unless stated otherwise. (Since the intensity isreal-valued and positive when the illumination is partially coherent,the tcc operator is Hermitian, and its eigenvalues are positive realnumbers.) When this normalization convention is followed, the tcc Mercerseries takes on the simple form seen in FIG. 1B, in which theeigenvalues are absorbed, and so do not appear explicitly. By choosingthe OCS kernels to be the eigenfunctions of the tcc kernel, and orderingthem in the equation of FIG. 1B such that the eigenvalue for each termis larger than the eigenvalues of all later terms (including thetruncated terms, i.e. the infinite set of terms that are omitted) theexpansion has the desirable property that each successive product termin FIG. 1B will represent the best possible coherent approximation tothe remaining portion of the tcc that preceding terms have not been ableto capture. However, it is known in the art that alternative coherentkernels may be chosen instead of the tcc eigenfunctions in order tobetter match the kernels to particular mask content of strategicinterest. These kernels are no longer optimal in the sense of bestmatching all portions of the tcc without regard to contentspecialization, and will likely not be entirely optimal even forspecialized mask content if they are chosen in a partly heuristic way.Nonetheless, even though the “O” in OCS stands for Optimal, the term OCSwill, for simplicity, sometimes be used herein to refer generally to theuse of any set of exclusively coherent systems to approximate apartially coherent tcc, even when the coherent system kernels are notoptimal. (The term “OCS” will not generally be applied when coherentsystems are used jointly with other kinds of systems.) It should also benoted that in a standard implementation where OCS aims to reproduce theTCC (or tcc) as a whole, the physical coherent systems that OCS defineswill approximately match all images produced by the physicallithographic system, even though these coherent systems need only besimulated (with adequate accuracy) in order to provide properdimensional compensation in masks. This matching at the physical levelis reflected in standard nomenclature, e.g. “Sum of Coherent Systems” or“Optimal Coherent Systems”.

It will be seen that the novel decomposition methods employed by theinvention likewise define physical systems whose superposed outputsmatch the physical lithographic system (though the matching systemsemployed by the invention include systems that are not coherent, and thematching accuracy obtained is superior to that from purely coherentsystems).

In the frequency domain, the Hopkins equation operates on pairs ofspatial frequencies, i.e. on pairs of plane waves that diffract from themask under illumination by mutually uncorrelated plane waves emitted bythe specified source. As is customary in computational lithography, theterm source refers to the shape of the illumination directionaldistribution, and a source is quantitatively specified by a map of theintensity of the illumination that is incident on the mask from eachdirection, with the intensity from any single direction being governedby the intensity of a single source point. Each illuminating plane waveproduces a coherent image contribution per the Abbe decomposition, butthe total image is partially coherent (despite the mutual incoherence ofthe separate illuminating waves), since the source shape only fills alimited portion of the full hemisphere of possible illuminatingdirections, in most cases only a very limited portion. When a given maskpattern is illuminated by a wave at normal incidence, the amplitude ofthe diffracted plane wave which has frequency f is written as M(f),where M(f) denotes the Fourier transform of m(x). The argument f alsodetermines the direction at which the diffracted order propagates awayfrom the mask, and f may be specified in direction cosine units. Whenthe illumination is shifted away from normal incidence to a direction s,the propagation direction for frequency f shifts to a direction f+s, butunder Hopkins imaging the amplitude is considered to remain M(f). Thoughnot shown explicitly, it should be understood that f usually denotes a2D spatial frequency that has x and y components. Each pair ofinterfering plane waves contributes an intensity modulation at thedifference frequency, and the integrated superposition of theseintensity harmonics gives the total image intensity, as shown in thefirst line of FIG. 1D. The OCS/Mercer kernels of TCC(f₁,f₂) are denotedΨ(f), and they are the Fourier transforms of the ψ(x) Mercer kernels oftcc(x₁,x₂). Per FIG. 1B, the ψ(x) are also the spatial-domain OCSkernels. When the frequency-domain TCC is expanded as a Mercer series,the image intensity becomes a sum of squared Fourier transforms of aseries of differently filtered copies of the mask spectrum M(f), witheach such filtering being imposed by the aperture of a coherent system,and with the Ψ(f) kernels serving as transmission functions of thesecoherent system apertures, i.e. as filter functions that modulate theM(f) spectrum, as shown in the 2nd line of FIG. 1D. The truncated Mercerexpansion of the TCC itself is shown in FIG. 1E. When the Ψ(f) aperturesare chosen to be the eigenfunctions of the TCC, the resulting OCS systemwill generally be a far more efficient decomposition set than the Abbeset of shifted apertures if practical levels of matching accuracy aresought, e.g. accuracies of order 0.1% to 1%.

Standard OCS kernels can be shown to inherently be smooth functions,based on the following argument: First, standard OCS kernels are chosento be optimal in a least squares sense (the “0” of OCS in fact standingfor Optimal), and it is known that an optimal kernel of this kind mustbe an eigenfunction of the tcc. It then follows from the mathematics ofeigendecomposition that the OCS kernels must (prior to their finalnormalization) be the particular functions of unit integratedtransmission that produce maximal intensity at the geometrical imagepoint when imaged according to the full tcc (with an added constraint onthis maximization that each new kernel must be orthogonal to allpreceding kernels). One consequence of this maximization property isthat any discontinuity in a putative ψ distribution would causeunnecessary light loss from diffraction beyond the lens aperture,implying that the ψ must instead be smooth functions. The same holdstrue for the frequency domain eigenfunctions Ψ(f), which are simply thediffraction spectra of the corresponding ψ. This follows becauselithographic sources are engineered to exclude illuminating directionswith greater inclination than the numerical aperture of the projectionlens (and in any case the source directional range is sharply bounded);thus any discontinuity in the Ψ spectrum entails the unnecessarydiffraction of light away from the geometrical image point duringpropagation from the exit pupil to the wafer. This continuity in thecoherent system aperture spectrum Ψ(f) stands in stark contrast to thefunctional form of the lithographic lens aperture; the circular physicalaperture of the latter lens constitutes a frequency-domain cutoff filterwhose edge sharpness is essentially infinite over all scales ofrelevance to computational lithography. It is true that, while smooth,an OCS kernel Ψ_(j)(f) will typically exhibit stronger and strongercurvature as the order index j is increased, in many cases approachingthe sharpness resolution of practical computational grids, butnonetheless a very large number of OCS terms will generally be needed inorder to closely account for the TCC-impact of the sharp lens aperture,due both to the smoothness of OCS kernels, and to the restrictivemathematical structure of Mercer terms, as will be discussed. As aresult, it will be seen that OCS's use of a Mercer series to approximatethe TCC usually entails a non-negligible accuracy tradeoff.

The linear convolutions in FIG. 1C are preferably carried out using fastFourier transforms (FFTs), and so execute rapidly. OPC is made practicalby the rapid speed of the FFTs in FIG. 1C of FIG. 1, but the criticalOPC accuracy/speed tradeoff is nonetheless gated by the number of FFTsemployed. FFTs use a discrete sampling or gridding, and while the exacttcc is strictly bandlimited (meaning that the frequency-domain TCC onlytakes on non-zero values within a finite region of thedoubly-dimensioned Hopkins domain), the approximate FIG. 1B truncatedexpression will include weak content outside the band limit. To avoidaliasing it is therefore desirable to use a spatial-domain gridding thatis finer than the theoretical Nyquist spacing defined by the lensresolution. In addition, phenomenological resist models are oftenemployed that include e.g. heuristic thresholding operations whichintroduce high-frequency content. With modern lithographic systemsoperating at for example NA=1.35 and λ=193 nm, one might typicallyemploy a sub-Nyquist spacing of about 10 nm.

Standard OCS kernels are derived by eigendecomposition of the fullfunctional map of the TCC. FIGS. 2A and B explain how the TCC value foreach pair of spatial frequencies in this mapping is primarily driven bythe basic optical consideration of whether or not the partially coherentlithographic optical system allows the plane wave orders associated withthe two frequencies to interfere together in the image, and so createintensity modulation at the difference frequency Δf≡f₁−f₂. Inlithography applications the calculation of the degree of modulationproduced by a pair of object spatial frequencies (denoted TCC(f₁,f₂)) isprimarily a matter of determining how many source points allow bothorders to be collected. Many other effects must also be taken intoaccount during a Hopkins TCC calculation (e.g., vector interference,aberrations, etc.), and the invention does so, but the basic summationof source point contributions during a Hopkins partial coherencecalculation is the main driver of the TCC value, and is of centralimportance to the functioning of the invention. As FIG. 2A illustrates,the pair of orders will generally be able to interfere in the image ifboth orders are captured by the lens. The ability of the lens to capturea given pair of orders will vary with illumination direction, and in theFIG. 2A example the illuminating plane wave is from a source point onaxis, and is seen in that case to allow both orders to be captured,whereas the obliquely incident plane wave considered in FIG. 2B doesnot, since the obliquity of the incident beam in the FIG. 2B case issufficient to shift the f₂ diffracted order outside the pupil. Note thatin FIGS. 2A and 2B the source is imaged to infinity and the illuminationfrom a single source point is thus effectively a plane wave. Also, itshould be understood that the detailed multi-element design oflithographic projection lenses will cause each collected wave to befocused to a point in the plane of the lens aperture, though forsimplicity this has not been shown in FIGS. 2A and 2B. Thus, even thougheach point in the source gives rise to a collimated bundle ofilluminating rays at the mask which are then diffracted as collimateddiffraction orders after propagation through the mask, each such orderwill either be entirely passed through to the image (if it focussesinside the sharp perimeter of the lens aperture), or it will fail inentirety to reach the image (if it focusses outside the sharp perimeterof the lens aperture). Also, it should be noted that even thoughcircular apertures are the most common pupil shape in lithographiclenses, other options such as annular apertures are also sometimesemployed. These apertures likewise have sharp perimeters, and theconsequences discussed herein of using circular apertures are alsogenerally applicable with these alternative apertures.

FIG. 2C illustrates how the TCC function for a specified pair of spatialfrequencies f₁ and f₂ can be understood physically to represent theoverall interference modulation produced by the two frequencies from allsource points. The construction in FIG. 2C that is used to determine theTCC value for frequencies f₁ and f₂ is known as the Hopkins diagram orHopkins construction. Rather than directly considering whether the f₁and f₂ diffraction orders both intersect the lens aperture when theirdiffracted direction is shifted by illumination from each of the varioussource directions (as was done in the examples of FIGS. 2A and B), theHopkins construction superposes on the frequency-domain source patterntwo copies of the circular pupil aperture that are shifted by amountsequal and opposite to f₁ and f₂ (i.e., equal and opposite to f₁ for onecopy of the circular aperture, and equal and opposite to f₂ for theother copy). When this construction is employed, f₁ and f₂ will both becollected when illuminated by source points in the regions labeled A,i.e. source points that lie within the intersection of the two shiftedpupil circles.

Thus, briefly stated, the main function of the Hopkins diagram isessentially to count the number of source points that succeed inproducing modulation. The source partial coherence is then accounted forby simply calculating the total source content within the intersectionarea (IA) shown in FIG. 2C, and normalizing that area by the totalsource area.

Of course, it will be clear to those skilled in the art that calculationof the TCC pursuant to the invention should preferably also take intoaccount such effects as defocus, aberration, wafer stack reflections,and vector imaging. As is now well-known, these effects may be handledby methods that modify the integration over the intersection area in theHopkins diagram. These methods are described, for example, in A. E.Rosenbluth et al., “Fast calculation of images for high numericalaperture lithography,” SPIE v. 5377 Optical Microlithography XVII(2004): p. 615. However, while these detailed imaging phenomena arepreferably included when integrating over the intersection area in theHopkins construction, it is the geometry of source point inclusion inthe Hopkins diagram that is of primary importance for the corefunctioning of the invention, and it is partial source inclusion withinthe Hopkins diagram that fundamentally defines the character ofpartially coherent illumination. Moreover, in standard photolithographypractice the Hopkins diagram or its mathematical equivalent is used todetermine the TCC function, and the TCC is then approximated for OPC asa sum of multiplied coherent OCS kernels, allowing the images producedby the partially coherent lithographic imaging system to beapproximately matched by a sum of coherent system images.

Referring to FIG. 3, the Hopkins TCC construction is now used toillustrate a limitation of the prior art methodology that has beendiscovered by the inventor, illustrating more particularly how the sharpedge of the lens pupil can cause slope-discontinuities in the TCCfunction at certain specific intensity harmonics. Theseslope-discontinuities are inherently difficult to approximate with asmall number of OCS kernels, and this will be seen to cause significanterror when using a truncated Mercer series of coherent systems toapproximate the TCC. In particular, when a truncated Mercer seriesapproximation to the exact TCC is subtracted from the exact TCC, theresidual portion of the TCC that the Mercer series fails to capture willgenerally be very large at near-DC intensity harmonics, as will bediscussed. The portion of the exact TCC which an approximate TCC failsto capture will be referred to as a TCC residual, denoted TCC^((r)). Thepresence of strong content in the TCC residual at near-DC intensityharmonics can be understood through study of FIG. 3, which shows theHopkins TCC construction being applied to a sequence of mask spatialfrequency pairs. More specifically, FIG. 3 depicts the changing TCCvalue implied by the Hopkins construction as one of a pair ofinterfering frequencies, namely f₁, is steadily decreased, while theother (f₂) is held fixed. Since the pupils in a Hopkins diagram arepositioned with a shift that is equal and opposite to the f₁ and f₂frequencies whose TCC is being calculated, FIG. 3 depicts the steadydecrease in f₁ as a rightward shift of the f₁ pupil aperture, with theHopkins construction for three successively smaller values of f₁ beingshown as one progresses from top to bottom in the figure. Initiallyf₁>f₂ (top diagram), which means that the difference frequency Δf≡f₁−f₂is positive. In this Δf>0 regime, the upper Hopkins construction in FIG.3 shows that the source fraction contained within the intersection ofthe f₁ and f₂ pupils is not changed by small rightward displacements ofthe f₁ pupil, since the f₁ pupil does not intersect the source under theexample conditions depicted in the upper diagram. This in turn meansthat the TCC holds constant in this regime, if for simplicity weconsider the case of aberration-free scalar imaging. As f₁ continues todecrease from its initial positive value, the DC condition of zerodifference frequency is reached, and the pupils in the diagram thencross as the difference frequency is driven through zero, as can be seenin the central Hopkins diagram of FIG. 3. At that point the pupil for f₁switches from being leftmost to being rightmost, and as a result thesource fraction contained within the pupil intersection begins suddenlyto decrease as f₁ is further decreased. This contained source fractionis essentially the TCC, and the sharpness of the pupil aperture causesthe onset of this substantial rate-of-change in the TCC to be completelyabrupt, i.e. to be arbitrarily sharp over scales of relevance incomputational lithography, giving rise to a slope-discontinuity in theTCC at Δf=0.

It can be noted with regard to FIG. 3 that the source portion that iswithin the intersection region could be outlined without shading, as inFIG. 2C, to emphasize that it is only this portion of the source whichcontributes to the TCC.

The slope-discontinuity that is illustrated in FIG. 3 is a specificexample of a behavior that occurs quite generally in the TCC, and thathas a significant impact on the accuracy of prior art OCS, as will bediscussed. It should be noted that while the difference frequencyΔf≡f₁−f₂ is steadily changed as f₁ is decreased in the FIG. 3 example,the mean frequency f≡(f₁+f₂)/2 is also steadily changing at the sametime, since f₂ is held fixed. However, the slope-discontinuity isgenerally a function only of the Δf change, not the change in f. Adirection or path of change in the frequency pairs which holds fconstant while changing Δf can be understood as a change along a rotatedor “slanted” coordinate, as will now be discussed. The TCC iscustomarily considered in the lithography field to be a function of thetwo mask frequencies f₁ and f₂, which are the coordinate axes of thedoubly-dimensioned Hopkins domain. However, the TCC can also be regardedas being a function of f and Δf, and these alternative coordinatesessentially constitute rotated coordinate axes in the Hopkins domain,i.e. axes which are not orthogonal to the f₁ and f₂ axes of the maskcontent. (It should be noted that a factor of ½ has been included in thedefinition of f in order that it represent a mean frequency, and thatthis ½ factor causes the distance metric to be different along the f andΔf axes. If the distance metrics are equalized, the f and Δf axes areseen to be rotated by 45° relative to the f₁ and f₂ axes.) When theeffect of steadily changing f while holding Δf at zero is depicted in aHopkins diagram, two exactly overlapped pupils are translated in unisonacross the source. Although the edges of lithographic source poles areoften illustrated as sharp, they are in practice actually slightlyblurred in the pupil plane. Moreover, while the source poles in the FIG.3 example have perimeters that partially parallel the edges of the lenspupil perimeter (though with a slightly different radius of curvature),the blurred boundaries between the on and off regions of most sourceswill generally have a substantially different orientation in the Hopkinsdiagram from the edges of the lens pupils, regardless of the f₁ and f₂values involved. For this reason the TCC generally does not change in asharply abrupt way when the frequency-pair is shifted along the axis ofthe rotated pupil coordinate f, i.e. the TCC exhibits relatively smoothchanges when f is varied with Δf held at zero, corresponding to atranslated pair of coincident pupil circles, as just described.Conversely, the TCC will generally include loci of sharp slope changewhen Δf is varied, as will be discussed.

FIG. 3 can be seen to convey that the TCC-contributing source portion(namely the source portion which is within the intersection region ofthe pupils) is neither increasing nor decreasing as long as the movingf₁ pupil is to the left (upper diagram), but that this contributingsource portion begins abruptly to steadily decrease in area once themoving f₁ pupil becomes the rightmost (middle and bottom diagrams).

Though FIG. 3 uses a simple f₁ sweep to illustrate a genericslope-discontinuity that arises in the TCC, it is the Δf component ofthe sweep that gives rise to the slope-discontinuity, and this will beshown to in turn impact the accuracy of the prior art OCS method. Ingeneral, TCC slope-discontinuities arise inherently from certainvariations along coordinate Δf, due to the abrupt change in the sign ofthe differential contribution to the TCC that occurs at all illuminatedperimeter portions of the (briefly aligned) sharp lens pupils when thetwo pupils cross in the Hopkins diagram.

Referring to FIG. 4, this can be understood as a consequence of theunique ability of the Δf=0 frequency pair to receive a contribution toits TCC value from those source points which diffract the particularmask spectrum component having spatial frequency f=f₁=f₂ exactly to thelens aperture. FIG. 4 includes inserts showing the lens pupil (note thatthey do not show the Hopkins construction); more specifically, the FIG.4 inserts show in highly schematic form the location in the pupil withwhich a particular example source point (not shown) diffracts differentorders f₁, f₂ into the lens pupil, in cases where f is the same in eachinsert, but where the Δf coordinate is different. The source pointchosen is one which diffracts the spatial frequency f exactly to thelens aperture when f₁=f₂, as depicted in the center insert. Since f₁ andf₂ are both (barely) collected for this Δf=0 frequency pair, the sourcepoint in question contributes to the TCC at Δf=0. On the other hand, theleft and right inserts illustrate that one frequency or the other of thefrequency-pair will fail to be collected when Δf is changed to any othervalue, i.e. to any non-zero value, meaning that the particular sourcepoint does not contribute to the TCC if Δf≠0, though it does contributeat the isolated Δf=0 location. Moreover, any other source point whichalready contributes at some finite Δf≠0 (for the same fixed f) will alsocontribute when Δf=0, and the TCC contribution from such a source pointabruptly switches on as soon as f₁ and f₂ are both collected. This onsetof non-zero TCC contribution is independent of the sign of Δf, since thecollection status of the pair remains unchanged if f₁ and f₂ areswapped; thus, a source point which begins contributing at some finiteΔf≠0 will continue to contribute over a finite range of differencefrequencies, including Δf=0. As the magnitude of Δf is increased ineither direction away from Δf=0, the Hopkins diagram pupils willsteadily sweep away an increasing number of source points that no longerachieve collection of both f₁ and f₂, causing a (locally) lineardecrease in the TCC in either direction away from Δf=0, with the sign ofthe change abruptly reversing as f₁ and f₂ interchange at Δf=0. A plotof the TCC as a function of small changes in Δf about Δf=0 willtherefore be very sharply peaked at the DC harmonic, as shownschematically in FIG. 4. This means that the derivative of the TCC withrespect to Δf will exhibit an arbitrarily steep jump at Δf=0 on anyscale of interest in computational lithography, i.e. the firstderivative will effectively be discontinuous, which means that thesecond derivative of the TCC with respect to Δf will effectively becomeinfinite at all values of f which are diffracted to the edge of the lenspupil by some portion of the source. The set of such f values constitutea locus of frequency pairs lying along Δf=0 where the exact TCC isslope-discontinuous (in the frequency domain). Such a locus will bereferred to as a “crease”. When the TCC is approximated using a finitenumber of OCS coherent kernels, i.e. with a truncated Mercer series, itis difficult for the finite set of Mercer terms to adequatelyapproximate the TCC creases, as will be discussed.

Qualitatively, FIG. 4 explains that the crease in the TCC arises from adiscontinuity that occurs in the sign of the contribution made by thosesource points which just diffract the fixed frequency f exactly to theedge of the lens pupil, with this discontinuity occurring when Δfreaches 0 during a sweep from e.g. negative to positive values, i.e.when the two pupils in the Hopkins diagram cross after they are movedtowards one another at equal and opposite speeds. Since Δf≡f₁−f₂, the Δfaxis along which the discontinuity occurs is slanted across the two maskdomains (f₁ and f₂) that form the doubled domain of the Hopkinsequation.

The discontinuity can be interpreted as an abrupt change in the sign ofthe differential contribution of source points at the edge of the twooverlapped pupils in a Hopkins diagram, and as a result it can becalculated in the Hopkins diagram as an integral around the rim of theoverlapped pupil circles. If aberrations and vector effects areneglected for purposes of discussion, the discontinuity can morespecifically be expressed mathematically by the equation of FIG. 5A inFIG. 5. In this equation S(f_(x);f_(y)) denotes the intensity of thesource as a function of pupil position. θ_(f) denotes the polar angle ofsource points along the edge of the coincident pupils (centered at f)relative to an axis along the Δf azimuth. The delta-function in FIG. 5Aarises from the slope discontinuity which occurs at Δf=0 (and does notconsider the merely finite curvature away from the slope discontinuity),and the magnitude of the second derivative of the TCC along thedifference direction Δf is therefore infinite at Δf=0, as indicated byFIG. 5B. In contrast, the second derivative in the orthogonal fdirection will be finite in magnitude as long as the source intensity Sis continuous, i.e. as long as the source directional pattern exhibitsits customary blur. An explicit expression for the 2nd derivative in thef direction is shown in FIG. 5A. Like FIG. 5A, FIG. 5C neglectsaberrations and vector imaging effects to aid clarity.

It should be noted that although the equations of FIGS. 5A and 5B arewritten in terms of a displacement along the f_(x) axis, the main termin the equation of FIG. 5A integrals is independent of the orientationof the coordinate axis (considering in this case only orientationswithin the 2D x,y space of the mask patterns). More specifically, thesecond line of FIG. 5A shows that the Δf azimuth choice only impacts themagnitude of the discontinuity through the |cos θ_(f)| factor. Ingeneral, the slope-discontinuity that constitutes the crease is presentacross all azimuths that have zero difference-frequency within the 2Dspace of the mask spatial frequencies. Though the equations herein oftensimply denote spatial frequencies as e.g. f for the sake of brevity, thespatial frequencies present in a 2D pattern actually have x and ycoordinates in general, and can also be understood to represent both thepitch and orientation of a phase oscillation, with FIG. 5A indicatingthat the 2nd derivative with respect to a difference frequency willexhibit a singularity regardless of the orientation of this differencefrequency within the x,y plane. Since the 4D TCC is very difficult todisplay and somewhat cumbersome to describe in equations, the presentinvention description will typically employ equations and illustrationsthat in appearance treat spatial frequencies as scalar variables along asingle coordinate axis, but it should be borne in mind that thesespatial frequencies are actually two-dimensional, and that the TCC is afour-dimensional quantity. It should also be noted that while thediscontinuity in TCC curvature is, to order of magnitude, broadlyindependent of orientation within the sub-manifold of the 2D x,y planeof the difference frequencies (which are the 2D frequencies of theintensity harmonics in the image), the extreme difference in magnitudebetween the infinite equation of FIG. 5A and the finite equation of FIG.5B shows that the TCC curvature has a very strong anisotropy within thefull 4D Hopkins space, and in particular between the f and Δf directionsthat, within this larger space, are rotated relative to the x,y plane.

FIG. 6 shows an example of the creases in the exact TCC function thatthe sharp pupil edge produces. The phenomenon is difficult to visualizeexcept with one dimensional (1D) spatial frequencies such as those usedin FIG. 6. Depicted is the TCC slice that is relevant to 1D patterns.Note that instead of plotting the TCC as a function of f₁ and f₂, FIG. 6uses rotated coordinates f≡(f₁+f₂)/2 and Δf≡(f₁−f₂). (Note further thatthe rotation referred to is carried out in the abstract doubled domainof the Hopkins integral, not in the physical space of the maskpatterns.) The spatial frequency units on the FIG. 6 axes are definedusing the “sigma” convention, which is commonly used in thephotolithography field, where the spatial frequency value assigned to amask modulation is the fractional position within the lens aperture stopwhere the diffracted modulation is focused when the mask is illuminatedat normal incidence. The dominant “crease” singularities occur when theorder interferes with itself (near-DC intensity frequencies) for sourcepoints that diffract the interfering orders near the edge of the pupil,so that the dominant crease (labeled in FIG. 6) is located alongportions of the f axis, where Δf=0.

Note that FIG. 6 is a simple example that plots a scalar TCC for a σ=0.5disk source at numerical aperture (NA)=0.8. This example is idealized inthat the disk source is treated as having a sharp perimeter. However, itwill be shown below that the Δf=0 crease is present even with realisticsources whose poles are blurred, and with sources of complex character,such as so-called free-form or SMO sources. It will further be shownthat the Δf=0 crease is particularly hard for a finite number ofconventional OCS/Mercer terms to reproduce. The FIG. 6 TCC may be seento have slope discontinuities at other frequencies besides Δf=0, and, aswill be discussed, there are embodiments of the invention that can beused to mitigate the impact of these additional discontinuities.

In computational terms, OCS kernels are used to approximate the TCCusing Mercer terms which take the form T_(OCS)=Ψ(f₁)Ψ*(f₂), as shown inFIG. 1E, i.e. each frequency-domain Mercer term is a product that isseparated in f₁ and f₂. The most common OCS practice is to choose thelargest yet-unused eigenfunction of the TCC as the Ψ function for eachnew T_(OCS) term in the series, in order that this new term provide thebest possible overall RMS fit to the as-yet-unfit portion of the TCC.(Here “largest” refers to the magnitude of the associated eigenvalue.)The creases in the TCC represent an effect whose behavior isdramatically different along the Δf and f directions, in that the secondderivative along the Δf axis is effectively infinite, while that along fhas a merely typical magnitude. The inherently smooth kernels in eachseparated OCS term are poorly suited to approximate this sharp crease,both because of their smoothness, and also because the Mercer terms havea relatively restrictive structure, as will be explained. Because ofthis limitation in the matching capability of coherent systems, residualerror in the OCS approximation will tend to “spike” near DC differencefrequencies.

Referring to FIG. 7, this difficulty in approximating the Δf=0 creaseusing coherent systems can be expressed mathematically using theequations of FIGS. 7A and 7B. Derived using a simple application of thechain rule, these equations show that when a Mercer term (i.e. T_(OCS))and its derivatives are evaluated at a pair of frequencies along thecrease of the TCC, the second derivative of the Mercer term along Δf hasa very similar form to the second derivative along f; these twoderivative expressions are seen to be composed of the same terms, withthe two expressions only differing in the signs given the terms (and bya numerical factor which simply reflects the use of a different distancemetric in the definitions of f and Δf). This similarity between theequations of FIGS. 7A and 7B reflects the somewhat constricted structureof a T_(OCS) term of the equation of FIG. 1B, which attempts to map the4D TCC^((r)) using a bilinear folding of a single 2D OCS kernel Ψ (or ψin the spatial domain). In physical terms, an OCS coherent systemattempts to match the behavior of a partially coherent system whosesource is typically of complex shape by employing the best possibleaperture pattern for the coherent system lens pupil (i.e. an optimal Ψ),but in general the former system will interfere pairs of nearly-equalfrequencies in ways that no coherent system can match. A secondlimitation involves the (identical) factors in FIGS. 7A and 7Bthemselves, which involve the OCS kernels and their first and secondderivatives. The OCS kernels are smooth functions, as previouslyexplained, so their derivatives tend not to have extremely largemagnitudes, particularly with Mercer expansions that are truncated forthe sake of compute time to include only the lower order OCS kernels. Itis therefore inherently difficult to develop large curvature differencesin the OCS-approximated TCC, given the form of the FIG. 7A and FIG. 7Bsecond derivatives, unless a very large number of Mercer terms areresorted to. On the other hand, the TCC exhibits second derivativesthat, per FIGS. 5A-5C, are essentially infinite along Δf while beingonly moderate in magnitude along f; thus the OCS/Mercer terms areinherently incompatible with such a behavior. After, e.g., 10 or 20Mercer terms have been employed, the OCS expansion will typicallyreproduce most portions of the TCC reasonably well, but significanterror will remain at the near-DC harmonics in proximity to the crease.It should be noted that even though OCS kernels are inherently smooth,higher-order OCS kernels will tend to have increasingly largercurvatures, and as more kernels are added the absolute error inresolving the crease will slowly decrease, though the relative error atthe crease will generally increase in comparison with the more rapidlydecreasing errors with which other parts of the TCC are rendered. Thus,as more kernels are added, the remaining error tends to becomeconcentrated in a narrowing vicinity around the crease region. This issomewhat analogous to the so-called “Gibbs phenomenon”, where the errorin approximating square-type waveforms using a finite number of smoothsine waves takes the form of a generic “ripple” that becomes a (largely)localized phenomenon arising within the vicinity of each squared-offedge discontinuity.

FIG. 8 shows the residual after Mercer terms formed from the first 11OCS kernels are subtracted from the TCC of the FIG. 6 example. In otherwords, the FIG. 8 surface plot shows the portion of the exact TCC thatthe prior art image calculation method fails to account for if 11 OCSkernels are used. Such an error kernel will be referred to as theresidual TCC, and may be denoted TCC^((r)) for short. As in the FIG. 8example, the term TCC^((r)) may refer to the portion of the exact TCCthat is left uncaptured by a standard truncated Mercer series, i.e. by aprior art OCS expansion, but we will also use the symbol TCC^((r)) torefer more generally to the error left behind by any approximation tothe exact TCC, including in some cases the TCC error that a specifiedinterim, incomplete, or partial set of series terms leaves behind. Inthe case of FIG. 8, the plotted TCC^((r)) is the result of subtractingN=11 T_(OCS) terms from the exact TCC discussed in connection with FIG.6 (disk source), with these T_(OCS) terms being formed (per FIG. 1E)from the 11 eigenfunctions of the exact TCC that have largesteigenvalue, as is standard practice.

As can be clearly seen in FIG. 8, the dominant feature in the uncapturedor residual TCC is the low-frequency (dose-like) “fin” near Δf=0. Thisrepresents a comparatively large error in representing the TCC in thevicinity of the crease, as is expected from FIGS. 7A and 7B. This isanalogous to the Gibbs phenomena in Fourier analysis. As with Gibbs, onecan expect asymptotically generic behavior in the residue once thekernel count becomes sufficiently large. The image impact of this finpartly resembles a dose error in its effect, since Δf=0 represents theDC harmonic. As discussed, this error can be understood as arising atfrequencies where the pupil circles in the Hopkins diagram approach acrossing condition. The fin is quite narrow but does have finite width,due to the finite curvature of the retained smooth OCS kernels. Thissmall but finite frequency span of the fin means that the intensityimpact will vary depending on the pattern content within each localregion of the mask, with the dominant scale length of the variationbeing several times wider than the projection lens resolution, butusually somewhat smaller than the simulation ambit, for example smallerby a factor of order 2, and thus several times smaller than the width ofa typical simulation frame. Since the variation tends to be gradual overscales comparable to the lens resolution, i.e. the intensity error tends(roughly speaking) to vary only over scales that are somewhat largerthan the typical individual features in today's IC patterns, it followsthat the impact is locally somewhat similar in qualitative terms to adose change; however this approximate dose variation can varysubstantially even within the confines of a single simulation frame(though enforced homogeneity in circuit content over e.g. micron scaleswill reduce this variation).

To increase accuracy when using a standard truncated OCS expansion it isnecessary to increase the number of systems (or kernels) N, but as N isincreased one finds that the remaining error tends to decrease moreslowly, so that the incremental accuracy advantage gained by inclusionof each successive OCS system faces diminishing returns. In addition,the TCC error that remains tends to become relatively more concentratedin the fin region, i.e. in the vicinity of the TCC crease.

Once a moderately large number N of OCS kernels have been employed, theapproximate TCC provided by FIG. 1E will generally show reasonableaccuracy in most respects, except that the sharp crease edges in theexact TCC will be rendered in the approximate TCC with excessiverounding, i.e. the rendered TCC will be rounded in the directionperpendicular to the crease.

FIG. 9A illustrates a second TCC example for 1D line/space features,where in this plot the TCC has been calculated with quite high accuracyby setting N to a value far larger than would be considered practicalwhen the equation of FIG. 1B is used during OPC; more specifically, Nhas been given the large value of 247. 247 OCS systems would normally beconsidered an adequate approximation to the exact TCC, but for reasonsof runtime efficiency N must typically be given a much smaller valueduring chip-scale IC applications. In this example the TCC is seen toexhibit the expected sharp “crease” along the DC contour where f₁=f₂,highlighted in FIG. 9A with a dashed line. Whereas FIG. 6 considered adisk source example, the non-limiting example of FIG. 9A considers thecase where a C-Quad Source (σ=0.6−0.95, 40° poles, Gaussian sourceblur), NA=1.35, xy-polarized, is used in the lithographic exposure. Thissource is shown in FIG. 9B, with the effect of Gaussian source blurbeing indicated schematically by the rounded corners of the poles.Vector imaging is assumed in the calculation, and the vector arrows inFIG. 9B show the polarization direction of different points in thesource. The spatial frequency axes use direction cosine units, includingthe coupling refractive index as a multiplying factor, in this caseequal to 1.44.

FIGS. 10A and 10B consider the same source and imaging conditions as areused in FIG. 9, and more specifically provide a comparison of close-upimages of the Δf=0 crease region when the approximate TCC is calculatedwith 24 and 247 kernels, respectively. It can be observed in the FIG.10A approximate plot that the bulk of the TCC is captured quite wellwith 24 kernels, but that the crease at Δf=0 is not. In particular, whenN=24 the crease is seen to be rendered with appreciable rounding. FIG.10B shows the same near-exact TCC as does FIG. 9, i.e. in both figuresthe plotted TCC slice for 1D patterns is obtained with 247 OCS kernels,but FIG. 10B uses the same perspective as the 24 kernel TCC plot of FIG.9A, which is oriented to highlight the crease. As discussed, N=247represents an impractically large kernel count for OPC applications, butthe TCC that it yields comes fairly close to the exact value, andcomparison of FIG. 9B and FIG. 9A makes it clear that the loss ofrendition accuracy in the latter N=24 case occurs primarily along therounded crease. Note too that even with the large value of N used inFIG. 10B, a small amount of rounding is still apparent in the renderedcrease. Though 247 kernels would normally be considered an adequateapproximation of the full TCC, the sharp crease cannot be exactlycaptured with any finite number of OCS systems.

The foregoing has shown that conventional Mercer terms cannot accuratelyrepresent the essentially infinite difference that (per FIGS. 5A and 5C)exists in the TCC between the second derivatives along Δf and f, withthis limitation arising from the structural similarity in these secondderivatives that a Mercer product of a smooth OCS kernel with itselfmust exhibit when the multiplied Ψ functions are separated along the f₁and f₂ axes, as has been demonstrated in FIGS. 7A and 7B. However, andreferring to FIG. 11, one can, in accordance with the embodiments ofthis invention, instead express the TCC or TCC^((r)) using a(non-Mercer) series whose terms will be referred to as “rotatedsystems”, denoted T_(Rotated). Each rotated system is the product of twodistinct kernel functions (instead of the single kernel Ψ used inOCS/Mercer terms) that are separated along the Δf and f axes (instead off₁ and f₂), as shown in FIG. 11A. Further, since Mercer terms arewell-suited for approximating most portions of the TCC, one morepreferred embodiment shown in FIG. 11A uses the T_(Rotated) terms torender the residual TCC rather than the full TCC, with the residual TCCbeing by construction the portion of the TCC that is comparativelyrecalcitrant to decomposition using a coherent system expansion (i.e. aMercer expansion).

The T_(Rotated) terms employ different kernels along the Δf and f axes,denoted {tilde over (T)} and

respectively, in order to readily capture the strong curvatureanisotropies arising at the crease. FIG. 11A thus provides an expansionthat is well-suited to approximating TCC^((r)). Of course, TCC^((r)) isexactly determined once a specified number N of OCS systems has beenemployed, but the goal of these expansions is to operationallyapproximate the TCC in a computationally efficient manner, since it isnot computationally feasible to use TCC^((r)) directly in imagecalculations at full-chip scale.

As has been discussed, each prior art coherent system can be appliedvery efficiently during OPC by means of FFTs, but close rendition of theTCC crease requires an inordinately large number of conventionalcoherent Mercer series terms. Practical use of an alternative series ofdecomposition systems not only requires that the new kind of systemaccurately render the crease, but also that the new terms involvekernels that are computable, and further that evaluation of the newterms for individual frames of mask data be sufficiently fast forfull-chip IC simulations. The question of accurate crease rendition willbe considered first.

To enable accurate rendition of the crease, the novel T_(Rotated)systems in FIG. 11A should in general use different functions for thetwo axes along which the system is separated, namely

along the f axis, and {tilde over (T)} along the Δf axis. FIG. 11approximates the residual portion of the TCC (e.g. the portion thatremains after application of N OCS terms) by additionally applying afinite number L of the new T_(Rotated) terms, with the different termsbeing distinguished using an index l. This recalcitrant residual TCCportion tends predominantly to be localized to the vicinity of thecrease, as has been seen in FIG. 8, where the resulting error takes onthe appearance of a “fin” when plotted, i.e. this error is present as anextended peak along the f axis that has a narrow cross-sectional widthalong the Δf axis. The relative importance of this fin region in theresidual error typically becomes more pronounced as more and more OCSkernels are resorted to in defining TCC^((r)). An expansion along therotated Δf and f axes is well-suited to represent this region since theTCC exhibits very different curvatures along these two directions, andthe use of two kernels (which can be given very distinct shapes) to formthe T_(Rotated) terms provides a direct way to render second derivativesof very different magnitude along the f and Δf axes, as may be seen inFIGS. 11B and 11C. In particular, if T is given a slope discontinuity atthe origin, it follows automatically from FIG. 11C that the secondderivative of T_(Rotated) with respect to Δf will become infinite atΔf=0. Similarly, FIG. 11B shows that the second derivative ofT_(Rotated) with respect to f will have an appropriately moderatemagnitude that will be shown capable of matching the finite secondderivative of TCC^((r)) along the f axis if the curvature of the chosen

function is given the appropriate moderate values at the various flocations along the crease. The infinite 2nd derivative in {tilde over(T)} simply means that the slope abruptly reverses sign at the origin,so that an appropriately constructed {tilde over (T)} function can beexpected to have a sharp peak or “tip” at the origin, while

will generally have finite curvature at all points. Such kernelfunctions enable T_(Rotated) to match the dramatically differentcurvatures that TCC^((r)) exhibits in the meridians parallel andperpendicular to the crease. (In intuitive terms, we expect

(f) to trace out the “ridgeline” of the fin peak, and {tilde over (T)}(Δf) to reproduce the average cross-section of the fin.) In contrast,each individual OCS system of the conventional Mercer series istypically only able to exhibit curvatures of broadly comparablemagnitude in the vicinity of the crease, as was shown in FIGS. 5A and5B, so that TCC^((r)) can only be well-matched by a conventional OCSexpansion if a large number of conventional coherent systems are used,which entails a significant cost in OPC runtime.

Referring to FIG. 12,

and {tilde over (T)} may be calculated explicitly as the particularfunctions which provide the best RMS fit to TCC^((r)) when used asT_(Rotated) factors. FIG. 12A shows in particular that such a fittingmay be made by minimizing the total squared fitting error, which isdenoted E_(Rotated). At its optimum, E_(Rotated) will exhibit nofirst-order change when small variations are introduced in

or {tilde over (T)}, where, in the case for example of

, the small variation may be assumed to take the form shown in FIG. 12B,in which a δ-function perturbation is introduced at an arbitrarylocation f′, with this perturbation having an infinitesimal complexamplitude c that may have arbitrary phase. When the perturbation shownin FIG. 12B is substituted into the equation of FIG. 12A and thevariation δE_(Rotated) extracted as a first-order quantity and set to 0,one obtains the variational condition shown in FIG. 12C. The equation ofFIG. 12C must hold for arbitrary complex-valued c, and this can only betrue if the optimality condition shown in FIG. 12D is satisfied at eachf′. The equation of FIG. 12D is not sufficient in itself for obtainingthe T_(Rotated) kernels, so we refer to the equation of FIG. 12D morespecifically as a first optimality condition.

When the decomposition is optimal E_(Rotated) must be minimized withrespect to small variations in {tilde over (T)} as well as

. Minimization with respect to {tilde over (T)} can be carried out usingsteps that closely parallel FIGS. 12B-12D, leading to a secondoptimality condition shown in FIG. 12E. If the equation of FIG. 12E isthen used to replace the {tilde over (T)}* factor on the right side ofFIG. 12D, we arrive at FIG. 12F, which allows

to be obtained explicitly. In particular, FIG. 12F shows that

is an eigenfunction of an operator Q(f′, f) that is quadratic inTCC^((r)), with Q being Hermitian in the doubled f domain. (Note thatthe residual TCC is not itself Hermitian when expressed in the rotatedcoordinates f and Δf.) FIG. 12F shows that the eigenvalues of Q aregiven by the product of the two factors in parentheses on the left sideof the first line of FIG. 12F. Since each eigenvalue is thus the productof the normalization integrals of the

and {tilde over (T)} {tilde over (T)} kernels, these normalizations areonly determined to within a product; this reflects the fact thatT_(Rotated) is unchanged by a complementary rescaling of both kernels.FIG. 12F allows explicit calculation of

, since standard and well-known methods are available to diagonalize theoperator Q. Moreover, as will be discussed, in a preferred embodiment ofthis invention only the dominant eigenfunction of Q needs to becalculated, and determination of an operator's dominant eigenfunction isknown to be particularly straightforward.

An operator equation for determining {tilde over (T)} can similarly beobtained by substituting from FIG. 12D into 12E. The result is FIG. 12G,which shows that {tilde over (T)} is the eigenfunction of the Hermitianoperator Z defined in FIG. 12G. Z can be seen to have the sameeigenvalues as Q. To finalize determination of

and {tilde over (T)} it is necessary to choose from among the differenteigenfunctions of Q and Z, and then to suitably normalize theeigenfunctions. If we substitute from the equation of FIG. 12D or 12Einto FIG. 12A, we arrive at the expression shown in FIG. 12H for theminimized value of E_(Rotated) obtained at the optimum. It follows fromthe first line of FIG. 12A that this residual squared error cannot benegative. FIG. 12H has the form of the difference between the integratedsquared residual TCC and the eigenvalue of Q and Z that is chosen whensolving for

and {tilde over (T)}. One can therefore conclude that to minimize theerror and achieve the best match one must choose the largest eigenvalueof Q and Z, i.e. that

and {tilde over (T)} should be chosen as the dominant eigenfunctions ofthese operators. Moreover, the process can be repeated by re-formingoperator TCC^((r)) after the newly obtained component has been removed,thereby enabling extraction of a new T_(Rotated) component. Ordinarilythis repetition would be equivalent to taking the second largesteigenelements of the previous Q and Z operators as defined from theoriginal TCC^((r)). However, as noted above and to be further discussedbelow, extraction of a T_(Rotated) component from TCC^((r)) is typicallyonly the first stage in a two-stage extraction of a loxicoherent system,and the loxicoherent system extraction that is actually carried out willbe different from extraction of the related T_(Rotated) component. Ingeneral, the eigenelements of Q and Z that are present after thedominant loxicoherent system is extracted from TCC^((r)) will not beequivalent to the higher order eigenelements in the pre-extractionversions of these operators.

To finalize determination of T_(Rotated) one chooses a normalization for

and {tilde over (T)}. According to FIGS. 12F and 12G, the eigenvalue inQ and Z of these kernels must be equal to the product of theirnormalization integrals, and this constraint leaves only a singleoverall free scale factor in their joint normalization. It is convenientto settle this floating factor by choosing {tilde over (T)}(0)=1.

FIGS. 12A-12H thus show that the T_(Rotated) systems are readilycomputed, and it has been shown previously that these systems arewell-suited to rendering the crease regions of the TCC that arerecalcitrant to standard OCS decomposition. However, it will be shownlater that the T_(Rotated) systems are not directly useable for fastimage calculations, but that, nonetheless, a more complex systemdecomposition can be derived from the T_(Rotated) systems that canaccomplish this function. In order to explain the steps of thisadditional system decomposition, it is necessary to first understand thegeneric behavior of the T_(Rotated) systems once a typically largenumber N of OCS systems is used to obtain the TCC^((r)) function fromwhich they are derived, such as N greater than about 10 or 20.

FIG. 13 shows a non-limiting example of the dominant T_(Rotated)separated kernels, obtained by applying the equations of FIGS. 12F and12G to the residual TCC that remains after N=24 OCS kernels have beenextracted from the C-quad TCC of FIG. 9, with FIG. 13A showing

and FIG. 13B {tilde over (T)}. Note that the horizontal scale of theFIG. 13 plots expresses the frequency arguments in so-called “directioncosine” units, which are proportional to the sine of the angle at whicha particular spatial frequency converges to the image when the mask isilluminated at normal incidence (so that the plotted direction cosinevalue of a collected spatial frequency can be larger than the numericalaperture of the lens if extended sources are used). In some cases thedirection cosine may, by convention, include the refractive index of thecoupling medium as a multiplying factor, facilitating comparison of thedirection cosine with the lens numerical aperture, but the FIG. 13 axesdo not include this multiplying factor. Frequencies f and Δf canalternatively be expressed in reciprocal-period units; in that case thefrequency that corresponds to a particular direction cosine value willbe equal to the direction cosine divided by the wavelength. In the FIG.13 non-limiting example the wavelength is 193 nm, and the coupling indexis 1.44.

The TCC^((r)) from which the FIG. 13 kernels were derived is shown inFIG. 14. The FIG. 14 example TCC^((r)) was obtained by subtracting 24OCS systems from the FIG. 9 TCC, i.e. N has been given the (typical)value of 24 when using a Mercer expansion to initially approximate theFIG. 9 TCC. FIG. 14 thus plots the error that would be incurred in usingthe standard OCS image approximation with a typical choice of systemcount, as well as depicting an input for the calculation of the FIG. 13rotated system kernels (since the post-OCS TCC^((r)) is such an input).The spatial frequency axes in FIG. 14 use direction cosine units,including the coupling refractive index of 1.44 as a multiplying factor.

The predominantly “fin-like” character of the residual TCC error canreadily be observed in FIG. 14. The fin represents the preponderantdifference between the approximate TCC shown in FIG. 10A, and the(almost) exact TCC shown in FIG. 10B. The fin is, in other words,essentially the deficit between the sharp crease in the FIG. 10B TCC(which is calculated using a very large number (N=247) of OCS kernels,and so essentially represents the exact TCC), and the rounded (N=24)crease rendition of FIG. 10A, with the sharp crease in the exact TCCgiving rise to the peak of the fin.

This fin-like predominant shape for TCC^((r)) is quasi-universal at thevalues of N typically used for OPC (e.g. N between 10 and 100), andrepresents a Gibbs-like phenomenon arising from the mismatch between theequations of FIGS. 5A-5C and FIGS. 7A and 7B. Additional TCC^((r))examples exhibiting this general behavior are shown in FIG. 15 for thecase of a so-called free-form or SMO source. The particular free-formsource that was used is shown in FIG. 15A. Sources of this kind tend todepart from the simple binary on/off intensity settings of moreconventional sources (though even conventional sources exhibitgradations in intensity at the edges of their illuminating regions, dueto the blurring with which they are rendered in the pupil), and FIG. 15Adepicts this varying intensity by using contours, with a singleheavy-line contour at 15% of peak intensity being used to demarcate therough boundaries of the illuminating source poles. Contours at 55% and85% are also shown (in a lighter line), corresponding to brighterintensities in the interiors of the poles, some of which have local peakintensities that are well below 100%. The source intensity at locationswell outside the 15% pole contours are substantially zero. The source ispolarized with an azimuthal orientation, as indicated by the insertedvector arrows in FIG. 15A.

FIG. 15B shows the residual TCC when N=24 OCS kernels are used toapproximate the TCC produced by the FIG. 15A free-form source whenprojected with NA=1.35 at λ=193 nm. The predominantly fin-likeconfiguration of the residual TCC is apparent. Although FIG. 15B doesnot attempt to depict the 4D TCC^((r)) that governs 2D patterns, the finis in fact present for such patterns, i.e. the fin is present as a peakwithin the 4D geometry of the full TCC^((r)), this higher-dimensionedpeak being very narrow in both dimensions of every 2D cross-sectionalmanifold, while being extended in the 2 orthogonal directions along the2D fin peak, the combined 4D configuration being of course difficult tovisualize. It should be emphasized that the 24 OCS kernels used tocalculate TCC^((r)) are the 24 most dominant 2D eigenfunctions of thefull 4D TCC (even though they are only applied to 1D patterns in thefigure), and are not merely eigenfunctions of the lower-dimensioned 2DTCC function that governs 1D imaging. This convention will be followedthroughout the description of this invention when quoting kernel counts,unless stated otherwise.

The TCC^((r)) fins shown in FIGS. 8, 14, and 15B represent the OCS errorwhen calculating the so-called “aerial image” that the projection systemproduces within the coupling medium before the IC wafer is interposedinto the imaging beam, i.e. the aerial image may roughly be regarded asthe incident image that is imposed on the wafer. However, OPC is todaytypically carried out using the actual exposing intensity within a planeat some designated depth in the resist layer, with this plane beingexposed by back-and-forth reflections of the image between the variousinterfaces of a wafer film stack in which the resist constitutes onlyone layer of many. For example, the table of FIG. 15C in FIG. 15 showsan exemplary wafer film stack in which layer #2 is the photoresistlayer. FIG. 15D shows the TCC^((r)) after 24 OCS kernels are used tocalculate the exposing intensity at the top of the resist layer (i.e. atthe interface between layers #1 and #2), under conditions where theprojection lens is focused at the midpoint of the layer, i.e. a rayintersection about 60 nm below the layer 1,2 interface. (The opticalfocus was set 93 nm below the upper surface of layer 1 to account forrefraction.) The combined aberrations caused by the film stack and bydefocus cause the TCC to become complex-valued, and FIG. 15D shows morespecifically the real part of TCC^((r)). The fin-like configuration ofthe residual error from the OCS approximation is again apparent, sincethe mechanisms that produce it are essentially independent of thepresence of a film stack, or of lens aberrations like defocus.

Lithographic exposure tools maintain very low levels of aberration,since modern IC requirements push resolution to the limit. Defocusrepresents a partial exception to this rule, because in practice itoften proves impossible to maintain a perfectly sharp focus acrossexposure fields of macroscopic dimension on heavily processed wafers.But even where defocus is concerned, the aberration levels that must betaken into account during mask design and OPC are generally well underone wavelength. Under such conditions of weak to moderate aberration thereal part of the TCC (and more specifically the real part of TCC^((r)))usually has a larger impact on image quality than does the imaginarypart, particularly with modern lithographic masks, which tend not to beof the hard-phase-shift type. A lesser impact from the imaginary part ofTCC^((r)) does not mean that the associated aberration itself will onlyhave a weak impact (though in practice this is likely to be the casewith aberrations other than defocus). This is because aberrations fromdefocus and the film-stack will generally cause the real part ofTCC^((r)) to increase, as well as producing a non-zero imaginary part.This aberration-induced increase in the real part of TCC^((r)) may beseen in a comparison of FIG. 15D (defocus and film-stack) to 15B(in-focus aerial image), bearing in mind the difference in verticalscales of the two plots.

However, the imaginary part of TCC^((r)) does impact the image, and theembodiments of this invention can achieve a significantly smallerTCC^((r)) in both the real and imaginary parts than can standard OCS atthe same compute budget, as will be discussed. FIG. 15E shows an exampleof the imaginary part of the residual TCC (denoted Im[TCC^((r))]), inthis case after extraction of 24 OCS kernels under the same imagingconditions as in the example of FIG. 15D. Like the real part ofTCC^((r)) (denoted Re[TCC^((r))]), Im[TCC^((r))] shows an increasedmagnitude near Δf=0, but an additional consideration comes into playwhere the imaginary part is concerned. In particular, the TCC^((r)) mustbe entirely real where Δf is exactly 0 (assuming that TCC^((r)) iscalculated following the extraction of purely OCS kernels), since theTCC is Hermitian. The near-DC “fin” in Im[TCC^((r))] therefore tends totake the form of paired peaks or “ripples” that have opposite sign, withthese “peak and valley” ripples closely flanking a zero-valued contourlying exactly along the Δf=0 axis.

FIG. 15F uses a contour plot to more clearly illustrate this genericconfiguration of the imaginary part of the fin, plotting the sameIm[TCC^((r))] example as the surface plot of FIG. 15D. Thin-linecontours are drawn at levels of +0.004 and −0.004, in order to encloselobes or “peaks” (and matching “valleys”) that have significantmagnitude in Im[TCC^((r))]. Note that even when these regions arereferred to as “peaks” for simplicity, the structure of Im[TCC^((r))]actually consists of paired regions that extend to (relatively) largemagnitudes in both positive and negative directions. To make thispairing clear the peaks that exhibit large negative magnitude are shownwith cross-hatching in the figure. The imaginary part of TCC^((r)) isseen to have anti-symmetry in Δf, so that Im[TCC^((r))] maintains itsmagnitude but reverses sign when mirrored about the f axis (which is thevertical axis in FIG. 15F). This is a consequence of the Hermitiansymmetry that the TCC must maintain (because the intensity that itprovides is always real-valued and positive). Im[TCC^((r))] is also seento have mirror anti-symmetry about the horizontal Δf axis of FIG. 15F;this reflects the bilateral symmetry of the FIG. 15A source about its xand y axes, operating in conjunction with the Hermitian symmetry ofTCC^((r)). OPC and mask design are usually carried out with symmetricsources, or at least sources that are symmetric by design, in order toavoid position shifts of the printed patterns in the presence of smallfocus errors. Even when OPC takes into account the small asymmetriesthat are actually present in measured sources, it is often quiteaccurate to neglect these asymmetries when considering the impact ofresidual error terms like Im[TCC^((r))] that are themselves alreadyquite small even in their predominant symmetric contribution.

In addition to the thin-line contours at height ±0.004 in FIG. 15F thatenclose lobes of non-negligible Im[TCC^((r))] magnitude, thick-linecontours at heights ±0.008, ±0.012, and ±0.016 are also included in thefigure. These higher contours show that the magnitude of Im[TCC^((r))]is fairly small except at locations near Δf=0, where Im[TCC^((r))]exhibits strong peaks and valleys at low difference frequency (eventhough the imaginary part is 0 where Δf is exactly 0). Except for theseparating split (i.e. a small anti-symmetric displacement) away fromthe f axis, the shape of the Im[TCC^((r))] fin peak as a function of fhas qualitative similarities to the shape of the real part (e.g.comparing FIG. 15E to the real part shown in FIG. 15D), allowing for achange in scale and the more complicated symmetry of the imaginary part.The error mechanism that predominates in the imaginary part of theresidual TCC after extraction of OCS kernels is the same as thatpreviously discussed in connection with the real part, and forconvenience we will continue to refer to the associated generic errorconfiguration in the imaginary part of TCC^((r)) as a “fin”, even thoughthat term is less well-suited to the anti-symmetric character of e.g.FIG. 15F than it is to the symmetric real part.

The methods provided by embodiments of this invention can substantiallycorrect the portions of Im[TCC^((r))] that are recalcitrant toextraction with OCS kernels, as will be discussed, but the method issimpler to apply to the real part of TCC^((r)), so non-OCS extraction ofRe[TCC^((r))] will be considered first.

T_(Rotated) kernels like those shown in FIG. 13 essentially constitute aclose model of the sharply peaked fin content in Re[TCC^((r))] after OCSextraction, with the sharp character of {tilde over (T)} at the originbeing particularly apparent in FIG. 13B. In lithographic applicationsthe real part of TCC^((r)) usually has an appreciably larger impact onthe image than does the imaginary part, and for simplicity thisdescription of the invention will generally use TCC^((r)) to refer tothe real part, unless otherwise stated. When {tilde over (T)} is griddedduring practical computations, the jump in the pixel-to-pixel valuedifference at the origin (involving a sign reversal) will decreaselinearly rather than quadratically with the fineness of the grid, makingthe second derivative of {tilde over (T)} effectively infinite at Δf=0on any scale of computational interest. In contrast, the secondderivative of

, though fairly large in magnitude at some values of f, will in generalalways be finite.

It should be noted that the central peak of the T factor in the dominantT_(Rotated) term of TCC^((r)) will almost always become quite narrowonce the number of OCS systems N assumes a value typical of currentpractice, e.g. larger than about 10. This reflects the stronglocalization of the OCS-recalcitrant portion of the TCC to values of Δfthat are near the Δf=0 crease. While the standard expansion in Mercerterms faces diminishing returns once the number of OCS terms reachesthis regime, the first terms of the FIG. 11A expansion will, incontrast, tend to converge very rapidly if N≥10 kernels have been usedto calculate TCC^((r)), since the T_(Rotated) factorization iswell-suited to capture the extreme difference in TCC curvature at thevicinity of the crease that then predominates in TCC^((r)). In short,the T_(Rotated) factorization is well suited to decomposing the parts ofthe TCC that are most recalcitrant to the standard Mercer decompositionused in current OPC practice, with such a Mercer decomposition beingused in the invention as well to generate TCC^((r)). In fact, when N hasbeen given a typically large value, the first l=1 T_(Rotated) term willitself usually represent a close approximation to TCC^((r)).

To exploit this behavior one can consider an idealized limit in whichTCC^((r)) (expressed along rotated coordinates) can almost exactly befit by a single rotated system, so that TCC^((r))(f,Δf)=

^((l))(f){tilde over (T)}^((l))(Δf) to a high degree of accuracy. Inthis limit one could fix Δf to any arbitrary value (denoted Δf_(fixed)),and then use the known values of TCC^((r)) to solve for

^((l))(f) to within a constant of proportionality (this proportionalityconstant being the reciprocal of {tilde over (T)}^((l))(Δf_(fixed))),i.e. we could set

^((l))(f)≡TCC^((r))(f,Δf_(fixed))/{tilde over (T)}^((l))(Δf_(fixed)).This idealized limit is not reached in practice, but there is in fact aclose correspondence between the l=1 T_(Rotated) term and TCC^((r)).Moreover, since the recalcitrant TCC^((r)) content that this T_(Rotated)term closely matches will be concentrated along the Δf=0 fin, we can inpractice determine

with quite high accuracy from the values that TCC^((r))(f,Δf) assumesalong the crucial Δf contour that traces the ridge of the fin, i.e. bychoosing Δf_(fixed)=0.

Of course, one might view such an approximation as unnecessary, sinceFIGS. 12A-12H provide exact values for the T_(Rotated) kernels. However,it proves helpful to consider an approximate (but generally quiteaccurate) calculation of the first T_(Rotated) kernel based on a fit atΔf_(fixed)=0, since it will be shown that such a fit is useful insubsequently decomposing the TCC in a way that allows fast calculationof optical images, which direct use of T_(Rotated) does not provide. Inpractice, such an approximate calculation could be made using a discretegridding of the TCC and kernels on a rotated {f,Δf} grid, so that thegridded values of the

(f) kernel would be approximately calculated as being proportional tothe value of TCC^((r)) at sampled f values along the Δf=Δf_(fixed)column of the rotated grid. Since the fin is centered at Δf=0, it ismost appropriate to choose the central Δf_(fixed)=0 column when makingsuch an approximate determination. Moreover, it will be shown thatachieving an accurate fit along the Δf=0 contour is particularlyimportant for accurate image calculation.

Referring to FIG. 16, FIG. 16A shows that the undetermined constant ofproportionality has the nominal value 1/{tilde over (T)}_(Δf=0) ⁽¹⁾. Thefirst part of the equation of FIG. 16B then shows that in the continuousdomain

(f) is approximately proportional to the value of TCC^((r)) along theΔf=0 axis (which is the fin peak), and without loss of generality we maydefine 1/{tilde over (T)}_(Δf=0) ⁽¹⁾ to have the value 1, since, asdiscussed above, the normalization scales of

and {tilde over (T)} are only defined to within a common shared factor(i.e.

can be rescaled by any arbitrary factor, so long as {tilde over (T)} isreduced by the same factor). FIG. 16B shows that

is then approximately given by the value of TCC^((r)) along the ridge ofthe fin.

To reiterate, though FIG. 12 provides an exact way to determine the

and {tilde over (T)} functions, such a determination only provides apartial solution to the problem of improving the speed/accuracy tradeoffduring image calculations. In particular, to obtain decompositionsystems that not only provide an accurate match to the TCC, but thatalso permit a computationally fast calculation of partially coherentimages, it is useful to further develop the FIG. 16B approximatesolution. Though

can be determined from TCC^((r)) at the fin peak using FIG. 16A, FIG.12E shows that {tilde over (T)} is impacted by other regions ofTCC^((r)), and FIG. 12E can in fact be used to optimally calculate{tilde over (T)} at all other values of Δf, including differencefrequencies that are quite far from the fin, thus accomplishing areduction in TCC^((r)) throughout the doubled domain. The firstT_(Rotated) term can then be applied with fair accuracy throughout theband limit of TCC^((r)) (assuming that N has been given a reasonablylarge value). However, with the exception of a lesser spike in TCC^((r))that is seen at the edge of the bandpass (e.g. at Δf≅2.7 in FIG. 14),which will be considered in a more sophisticated embodiment, TCC^((r))is generally small at locations away from the fin. The small value ofTCC^((r)) in these non-fin regions facilitates matching by the rotatedsystem since FIG. 12E will then give {tilde over (T)} a suitably smallvalue. This means that T_(Rotated) is critically determined by thebehavior of TCC^((r)) in the narrow region near Δf=0, where thedeficiency from neglected OCS terms beyond the cutoff N is significantlylocalized. Moreover, it will be shown below that the lesser TCC^((r))spike at the bandpass edge is a distinct localized behavior which,though it arises from a (weaker) discontinuity associated with the sharppupil edge, is completely independent of the dominant slopediscontinuity that gives rise to the fin.

This means that once N has been given an adequately large value, thefirst T_(Rotated) term (l=1 in the FIG. 11A expansion) essentiallyexpresses the behavior of the intrinsically emergent fin, and so can beused to quantitatively express a generic behavior exhibited by theresidual TCC. This generic behavior may be regarded as a Gibbs-likephenomenon, in the sense that it is an upswing in error that istriggered at all frequency pairs which involve a particulardiscontinuity (here a slope-discontinuity), and is largely localized tothe vicinity of the discontinuity, and further because it results from adeficit of higher-order terms in an infinite series expansion that isill-suited to rendering the discontinuity. The association of therotated system kernels with the fin may be understood more specificallywith reference to FIG. 17, which indicates schematically the role of the

and {tilde over (T)} factors in approximating the residual TCC, usingthe disk-source TCC^((r)) that was shown previously in FIG. 8 as anon-limiting example. When TCC^((r)) is modeled with the expressionT_(Rotated)=

(f){tilde over (T)}(Δf), the

term essentially represents the frequency-dependence of the overallmagnitude of the fin-like deficit in matching the full TCC, and per FIG.16B this function will approximately be given by the value of TCC^((r))along the ridge of the fin, as depicted in FIG. 17A. The TCC deficitwill be concentrated within a narrow vicinity of the fin, which thefirst T_(Rotated) term approximates with what may be regarded as ageneric cross-section factor {tilde over (T)}(Δf), as shown in FIG. 17B.

Upon further decomposition, to be discussed, each T_(Rotated) term willbe found to yield a new system which will be referred to as a“loxicoherent system”, with this loxicoherent system consisting of apaired coherent system and incoherent system operating in sequence, with

being decomposed into the first system of the pair, namely a coherentsystem represented computationally by a bilinear product of coherentaperture functions which will be referred to as mask filters, denotedT′, and with each loxicoherent system further comprising a second systemwhich is an incoherent system, represented computationally by what willbe referred to as an intensity filter, or intensity kernel, orincoherent kernel (these terms being synonymous), denoted T″, with thisintensity kernel being essentially a revised version of {tilde over (T)}that typically will no longer follow FIG. 12G exactly. Much like thefirst rotated system, the lowest order or primary loxicoherent systemcontinues to reproduce the quantitative details of the generic localizedfin behavior, as well as providing a further partial reduction ofTCC^((r)) over the entirety of the Hopkins double domain.

It should be noted that this generic behavior differs in its detailsfrom that of the classical Gibbs phenomenon. The latter phenomenon givesrise to a generic localized ripple pattern of quasi-fixed characterthat, in the inverse-transform domain (e.g. the spatial or time domain),is seen to arise at the vicinity of every sharp edge in a square-typewave, when that wave is reproduced with a truncated series in theFourier domain, i.e. a Fourier series from which all high orders havebeen removed. In contrast, when a truncated OCS (i.e. Mercer) series isused to approximately match partially coherent optical images, theassociated localized Gibbs-like behavior arises within the frequencydomain (i.e. near the creases in the frequency-domain rendition of theTCC) rather than in the inverse-transform domain, and its quasi-fixedcharacter is reflected in the fact that we can approximately reproducethe deficiency arising from the rounded rendering of the crease bymaking use of only a single {tilde over (T)} function of fixed shape.While the presence of a weaker spike in TCC^((r)) near the bandpass edgetends to reduce the accuracy of the approximation made in the equationof FIG. 16B, it will be shown that this weaker spike can largely beextracted from TCC^((r)) using a separately fitted system, making theequation of FIG. 16B a very accurate approximation for the stronglycrease-localized remainder that then results (with an analog of FIG. 16Bbeing applied first in a preferred embodiment, as will be discussed).

Another conclusion that can be drawn from this behavior is that eventhough the {tilde over (T)} kernel is determined from

as an optimized quantity (via FIG. 12E), it nonetheless only provides akind of averaged approximation to the fin cross-section (see FIG. 17B),which at finite N will not exhibit perfectly generic behavior. It willbe seen that control of the cross-sectional approximation should beconsidered when applying the invention to the 4D TCC^((r)). Conversely,even though

is only determined in an approximate way when FIG. 16B is used, itnonetheless exactly captures the behavior of the key fin peak, asillustrated in FIG. 17A.

In terms of accuracy and kernel computability, the T_(Rotated) expansionof FIG. 11A provides a successful complement to the Mercer expansion ofFIG. 1E, in the sense that the two expansions are well-suited to addressthe totality of the TCC. (The weaker discontinuity at the Δf bandedgemay require a third system to closely extract, as will be discussed, butthis discontinuity is less resistant to extraction by OCS than is thedominant Δf=0 fin.) However, the FIG. 11A form of the T_(Rotated)expansion falls far short of the Mercer expansion in a key way, sinceFIG. 11A provides no immediate analog of the efficient image calculationformulae in FIGS. 1C and 1D. In other words, while inclusion of termsfrom a rotated decomposition would provide an accurate approximation ofTCC^((r)) due to their efficient capture of the fin residual that isrecalcitrant to approximation with the prior art coherent kernels, suchterms do not directly provide computational utility for OPC, since FIG.11A offers no increase in computational efficiency over the basicHopkins integral shown in the equation of FIG. 1A of FIG. 1. It is theFIG. 1C reduction of the Hopkins equation to a series of fastconvolutions that provides the OCS decomposition method with utility forsemiconductor manufacture.

It will now be shown that the FIG. 11A expansion is the first part of afull decomposition into what will be referred to as loxicoherentsystems, where these loxicoherent systems provide a key practicaladvantage over the T_(Rotated) expansion by enabling a very efficientimage calculation step. As with rotated systems and coherent systems,the loxicoherent systems are formed from kernels whose arguments are oflower dimension than the TCC being decomposed, i.e. 2D when the TCC is4D. Like the rotated systems (but unlike the prior art coherentsystems), each loxicoherent system is characterized computationally bymore than one distinct kernel (for example, two distinct kernels in thesimplest embodiments, where these two kernels characterize the twodifferent apertures of a paired coherent system and incoherent systemthat operate in sequence), and in most embodiments the argument of atleast one of these kernels is separated along an axis that is notorthogonal to the primary f₁ and f₂ mask content axes that form thedoubled Hopkins domain.

It will further be shown that the first term of the full loxicoherentdecomposition usually captures virtually all of the large increase inTCC rendition accuracy that the first T_(Rotated) term provides. It willalso be shown that when the T_(Rotated) decomposition is only the firstpart of a full decomposition into loxicoherent systems, it is preferablein lithographic applications to calculate the first term of the newexpansion using FIG. 16B, rather than FIGS. 12F and 12G.

Referring to FIG. 20, the second part of the full decomposition intoloxicoherent systems is carried out computationally by decomposing the

(f) factor in each T_(Rotated) term. This will be shown to provide thelens aperture transmission that defines the first constituent pairedsystem of a loxicoherent system, namely the constituent coherent system.(Here the first part of the full decomposition refers to thedecomposition of TCC^((r)) into T_(Rotated). As will be discussed, thesecond part of the decomposition initially provides the first coherentsystem in the paired sequence of constituent systems that form theloxicoherent system, with the final step of the decomposition then beingthe determination of the incoherent system of the paired sequence.) Inparticular,

(f), which can be regarded as a function of f₁ and f₂ via f≡(f₁+f₂)/2,is decomposed into terms that are separated along f₁ and f₂ as bilinearproducts of a new kernel function denoted T′, as shown in FIG. 20A,where FIG. 20A shows how the second part of the loxicoherent systemdecomposition is carried out on the jth T_(Rotated) term. Thedecomposition of

(f) yields at least one coherent system, i.e. this decomposition has thesame computational form as a Mercer expansion, meaning that thisdecomposition could in principle be carried out using aneigendecomposition of

(f) (considered as a function of the two arguments f₁ and f₂), e.g., onecould carry out an eigendecomposition of a matrix that expressed agridded sampling of

(f) along f₁ and f₂ (which for 1D kernels would be a Hankel matrix, towhich specialized eigendecomposition methods may be employed). However,while such a procedure would be appropriate in a context where adecomposition of a TCC into purely coherent systems was sought, theembodiments of this invention make use of novel loxicoherent systems,which in a preferred embodiment each comprise a sequentially pairedcoherent and incoherent system. A proper decomposition of

(f) into coherent systems for pairings of this kind is quite efficient,as will be discussed; i.e. a preferred means for decomposing

(f) will be shown to converge very rapidly in the retained terms of thefull loxicoherent systems. In contrast, an independent decompositioninto coherent systems (in particular, an eigendecomposition) of anoperator like

that has Hankel form will generally be slow to converge, and whencarrying out a loxicoherent decomposition it is therefore preferable totake into account the presence of the {tilde over (T)} factor, which, aswill be discussed, has a very strong impact on the optimal choice of theT′ kernels. This means that even though the decomposition of

[(f₁+f₂)/2] into at least one coherent system has the form of atruncated Mercer series, the most efficient choice for the terms of thisseries are not, in general, the usual eigenfunctions of

, and in fact will usually depart very substantially from theseeigenfunctions. In carrying out the decomposition of

′ a specific number K of separated terms is kept. When j is 1 in FIG.20A it is usually preferable to set K to 1, so that

is decomposed as a single coherent system, with kernel T′. The fact thatthe interaction of the T′ kernel with {tilde over (T)} will generallycause the optimal T′ kernel to depart very substantially from anyeigenfunction of

is related to the fact that the constituent coherent and incoherentsystems of a loxicoherent system would ordinarily provide an extremelypoor fit to TCC^((r)) if considered individually. It is only as asequentially operating pair that the constituent systems togetherprovide a close match.

Each term on the right side of FIG. 20A forms the basis of aloxicoherent system. In certain preferred embodiments only a limitednumber of loxicoherent systems (in most cases a single such system) areextracted from each specific residual TCC. If additional loxicoherentsystems are desired, it is preferable to first carry out a newT_(Rotated) expansion on the remaining (i.e. updated) residual TCC, andthen to extract further loxicoherent systems from the new T_(Rotated)expansion of the updated TCC^((r)). In these embodiments eachloxicoherent system may be chosen so as to maximally reduce the residualTCC, or to maximally reduce a specified sector of the residual TCC, aswill be discussed. In general (but with some exceptions to bediscussed), each new loxicoherent system that is deployed will bedesigned to at least strongly reduce the particular residual TCC thathas been left unextracted by application of the preceding coherent andloxicoherent systems. Toward that end, the {tilde over (T)}(Δf) factormay be individually revised within each system in order to maximize thereduction of TCC^((r)). This is indicated notationally in FIG. 20B ofFIG. 20, where the expression therein provided for the lth loxicoherentsystem includes a new kernel T″(Δf) in place of the T(Δf) kernel thatappears in the right-side of FIG. 20A, reflecting the fact that theT″(Δf) kernel may be adjusted away from T(Δf) in a way that improves thematching to TCC^((r)), as will be discussed. This adjustment typicallyrepresents the final step in the extraction of a loxicoherent systemfrom TCC^((r)). Note that in the equation of FIG. 20B and subsequentequations the various loxicoherent system terms will be distinguished bya single index such as l, even though these terms may arise from anesting of series in the first and second parts of the loxicoherentsystem decomposition, e.g. from nesting a T_(Rotated) system seriesextraction indexed by j and a Mercer-form series expansion of

([f₁+f₁]/2) indexed by k. The single l index simply enumerates thevarious j,k combinations that arise in applying FIG. 20A. It should alsobe noted that the two-part decomposition of a residual TCC intoloxicoherent systems is, in a preferred embodiment, a two-part procedurefor extracting a loxicoherent system from a residual TCC, with thisprocedure then optionally being repeated to extract successive newloxicoherent systems from the residual TCC left behind by each precedingextraction.

Another point to note is that there are embodiments of the invention, tobe discussed in detail, in which the separate T″ kernels in a pluralityof loxicoherent systems are jointly optimized together, in such a waythat the loxicoherent systems collectively extract the maximum possibletotal portion of TCC^((r)), rather than each T″ being optimized as asingle constituent kernel function to maximize the extraction providedby its own system.

A key aspect of the loxicoherent system is that it is formed from atleast two distinct constituent lens system apertures, representedcomputationally by at least two distinct kernel functions that specifyconstituent lens aperture transmissions (or their autocorrelations),such as T′ and T″, unlike the prior art coherent systems used in OCS,which use a single coherent lens aperture with transmission Ψ, and soare formed computationally by multiplying two copies of the same kernelfunction Ψ. In the FIG. 20B embodiment each loxicoherent system consistsof two constituent systems, the first being coherent and the secondincoherent, and the loxicoherent system is represented computationallyin FIG. 20B using two kernels; T′ is a mask filter specifying the lensaperture transmission of the constituent coherent system, and T″ is anintensity kernel specifying the autocorrelation of the lens aperturetransmission of the constituent incoherent system, with T″ also beingreferred to as a dose kernel or incoherent kernel. The Fouriertransforms of T′ and T″ are spatial-domain functions, denoted t′ and t″.The combined right-side term of FIG. 20B describes the contribution madeby the loxicoherent system in matching the TCC, and for simplicity theright-side of the equation of FIG. 20B can itself be referred to as aloxicoherent system. The roles of T′ as the aperture transmission of aconstituent coherent system and T″ as the autocorrelation of aconstituent incoherent system aperture will be shown to followimmediately from the mathematical structure of a loxicoherent systemmatch to TCC^((r)).

It will now be shown that these loxicoherent systems differ fromcoherent systems in a number of ways; for example they employ twoconstituent imaging stages operating in sequence, with the first ofthese stages (a coherent system) using the amplitude transmitted by themask object (coherently illuminated) as input, but with the secondsystem of the sequence (an incoherent system) using, as an incoherentinput, the intensity generated by the image that is output from thefirst constituent system. The embodiments of this invention then computethe output of the incoherent system, and use this output as one(loxicoherent) contribution to the intensity with which the partiallycoherent image intensity from the lithographic system is approximatelymatched, further using this matching intensity to process the incomingframes of mask data, as will be discussed.

In accordance with well-known theory, each computation of an incoherentintensity is made by convolving the input intensity pattern (which inthis case is the image intensity produced by the paired coherent system)with a kernel (e.g. t″) that represents the squared inverse Fouriertransform of the transmission of the incoherent system lens aperture,which is mathematically equivalent to the inverse transform of theautocorrelation of the incoherent system aperture. Only the computedintensity is required, but in physical terms the output image from thecoherent system becomes incoherent if it passes through a fine diffuser,or if it excites self-luminous emission from the image-plane medium(which is the object-plane medium for the incoherent constituentsystem). When employed in accordance with the invention the loxicoherentkernels provide a computationally fast and comparatively accurateestimate of the image contribution made by the residual TCC (TCC^((r))),with this computational speed benefit successfully overcoming a criticallimitation of the T_(Rotated) systems. The conventional approach ofaccounting for the residual image contribution by increasing the numberof coherent systems N in an OCS expansion will be shown to requiresignificantly more computation than do loxicoherent systems in order toattain a comparable accuracy. It will also be shown that in many casesone can efficiently gain a further improvement in accuracy bysupplementing the first loxicoherent system with what will be termed aDC-monolinear system.

Equations 8C through 8I relate to the efficient calculation ofcontributions made by loxicoherent systems in matching partiallycoherent images. The intensity error ΔI^((r))(x) from truncating the OCSexpansion with N coherent systems is given by FIG. 20C, which has thesame form as the typical frequency-domain Hopkins equation (FIG. 1D),but with the full TCC being replaced by the residual unaccounted-forportion of the TCC, i.e., by TCC^((r)). In one preferred embodiment eachsuccessive loxicoherent system is chosen to strongly match the remainingTCC^((r)), so the intensity error ΔI^((r))(x) is well approximated byreplacing TCC^((r)) in FIG. 20C with the equation of FIG. 20B, to obtainFIG. 20D. The structure of FIG. 20D shows that the intensitycontribution provided by the lth loxicoherent system results from anintegration over the doubled Hopkins domain of the mask, in this casethe doubled frequency domain consisting of all pairs of mask amplitudefrequencies f₁ and f₂, with each amplitude frequency being passedthrough a coherent system lens aperture that has transmission T′, thusproducing an intensity modulation component when the amplitudes at thetwo frequencies interfere after being transmitted by the constituentcoherent system to the coherent system image plane, this interferenceintensity component being modulated at the difference frequency f₁−f₂.The constituent incoherent system then transmits this intensitymodulation to the output of the loxicoherent system, with the intensitytransmission being given by T″_(l)(f₁−f₂). In the image plane thisintensity modulation oscillates spatially, as specified by the factore^(2πi(f) ¹ ^(−f) ² ^()x).

The paired coherent and incoherent constituent systems operate insequence, so that computationally the coherent system is accounted forfirst. If one applies the T′ mask filter to the mask spectrum M (i.e. tothe Fourier transform of the mask patterns), thereby representingcomputationally the transmission of the mask amplitude through theconstituent coherent system of the loxicoherent system (in this case thelth loxicoherent system), one obtains the filtered version M′ of themask spectrum defined by FIG. 20E. By substituting M′ into the equationof FIG. 20D, one obtains, after switching the variables of integrationto the rotated axes f and Δf, the FIG. 20F expression for the imagecontribution provided by the lth loxicoherent system. FIG. 20F ispartially separated in the rotated variables, making the f integrationequivalent to an autocorrelation of the filtered mask spectrum, thusallowing the equation of FIG. 20F to be written in the form of FIG. 20G,where the star denotes autocorrelation.

By applying Fourier identities to the FIG. 20G frequency domainexpression, and then taking the inverse transforms of the right sides ofFIGS. 20G and 20E, FIG. 20H is obtained. FIG. 20H describesmathematically a new form of image decomposition, with the second termof FIG. 20H being the loxicoherent contribution, or more precisely thecontribution from the employed set of loxicoherent systems, which are Lin number. This second term is a sum over the inverse transforms of eachFIG. 20G term. In FIG. 20H t′ and t″ denote the inverse transforms of T′and T″. The N terms in the first summation of FIG. 20H representstandard coherent system terms, namely the OCS terms whose residualintensity error ΔI^((r))(x) is defined by the equation of FIG. 20C,while the L terms in the second summation provide the contributions fromthe new loxicoherent systems that closely match ΔI^((r))(x). Eachconvolution in FIG. 20H can be approximated by Fast Fourier Transform onthe usual sub-Nyquist sampling grids used in computational lithography,providing near-linear area scaling in the overall image calculation.

Two convolutions over the simulation field are used to obtain eachloxicoherent system contribution in FIG. 20H, whereas a prior artcoherent system requires only a single convolution. However, when L issmall the two additional convolutions in each loxicoherent system termof FIG. 20H provide a much greater accuracy improvement than would beobtained by adding two additional OCS kernels to the standard OCSexpansion, as will be shown. It should be noted that these conclusionsgenerally apply to two-dimensional mask patterns, i.e., mask patternswhich are functions of both x and y coordinates, even though forsimplicity the FIG. 20 equations only refer to a single mask coordinatex. Certain considerations particular to 2D patterns are discussed below.

The left summation in FIG. 20H uses coherent systems, and the lensapertures of these coherent systems (defined by their transmissionfunctions Ψ(f), or the inverse Fourier transforms ψ(x) of thesetransmission functions) can be chosen by prior art methods, e.g., as theTCC eigenfunctions used in standard OCS. The set of coherent systemsemployed by the invention will be referred to as the coherent systemset.

In preferred embodiments the number of systems N in the coherent systemset will be of the same order of magnitude as the number of coherentsystems employed by prior art OCS, e.g., N will generally be in therange of 10 to 100. However, it is possible in principle for N to be 0.Moreover, from a fundamental point of view one can readily constructcomputationally a loxicoherent system that will match the behavior ofany given coherent system (such as one of the coherent systems in theleft summation of FIG. 20H), since the constituent incoherent system ofa loxicoherent system can be given a fully open and corrected numericalaperture that is much larger than that of the constituent coherentsystem, i.e. the constituent incoherent system can be given a muchhigher resolution than the constituent coherent system, so that theintensity produced by the constituent coherent system is essentiallytransferred to the loxicoherent system output without further change. Inthat sense any coherent system could be regarded as merely aspecial-case loxicoherent system, and thus all of the imagedecomposition systems employed by the invention could be said to beloxicoherent systems, including those of the coherent system set.However, it would clearly be inefficient to actually expend computeresources on convolution with incoherent system kernels that merelyproduce a duplicative transfer, and the use of purely coherent systemsis well-known in the art. Since the loxicoherent systems employed by theinvention are novel, this invention description will continue todistinguish between the purely coherent systems of the coherent systemset and the irreducibly loxicoherent systems of the loxicoherent systemset. However, it should be understood that there are embodiments of theinvention which do not use coherent system sets per se.

The intensity contribution from each loxicoherent system (e.g. in theright summation of FIG. 20H) should be real-valued, and in a preferredembodiment each t″ spatial domain incoherent kernel is likewisereal-valued. Although the total intensity must be non-negative, it ispossible for individual loxicoherent systems in the expansion to makenegative contributions, and since the t′ convolutions are squared thesenegative contributions will arise from strongly negative regions in t″.As will be discussed, it is commonly the case that the predominantresidual content in TCC^((r)) shows a rapid dependence on Δf, but only amore gradual variation along f, i.e., the typical content of TCC^((r))tends to be laid out in “ripples” that lie along contours of Δf. Thisis, of course, strongly true of the pronounced fin along Δf=0 thatdominates TCC^((r)) before extraction of the first loxicoherent system,but it generally remains true to a lesser degree after the firstloxicoherent system has been extracted, i.e., when subsequentloxicoherent systems are extracted. These remaining “ripples” or ridgeswill be considerably weaker than the removed fin along Δf=0, and if l≥2there will generally be no single remaining ripple that predominatesover the other ripples to the same degree that the Δf=0 fin did beforebeing extracted by the l=1 loxicoherent system. For this reason we mayrefer to the first loxicoherent system as the “primary loxicoherentsystem”, or sometimes as the “first-order loxicoherent system”.

If the sign of TCC^((r)) in the frequency domain can be approximated asbeing only a function of Δf and not of f, it can be advantageous toencode the sign of the loxicoherent contribution in the sign taken on byT″(Δf) as Δf changes. However, the T′ kernels are functions of f₁ and f₂in the loxicoherent system, not f, and as a result it will usually notbe possible to have a fully consistent sign encoding across multipleridges, making it eventually desirable to resort to multiple additionalloxicoherent systems. A savings in compute time is nonetheless possiblewhen these multiple ridges have a predominantly “diagonal” or “slanted”orientation along contours of Δf, since this allows the specialized formof the loxicoherent expansion shown in FIG. 20I to be employed, whereinsome coherent system inputs (which may be the input from a singlecoherent system, if R_(l) ⁽⁺⁾ is 1) to the lth incoherent system aregiven a positive sign, while other coherent system inputs (or singleinput, if R_(l) ⁽⁻⁾ is 1) to the lth incoherent system contribute with anegative sign. (Here “diagonal” and “slanted” should be understood asreferring to a concentration along f₁−f₂ contours due to previousrelative exhaustion by OCS of content that is well-aligned with f₁ orf₂, even though “diagonal” might arguably be appropriate in a literalsense only for 1D patterns.) FIG. 20I can be regarded as a special caseof FIG. 20H in which the t″ kernel for some systems is equal andopposite to the t″ kernel of other systems, allowing them to be groupedas in FIG. 20I. When such a pairing is enforced explicitly, as in FIG.20I, one need only carry out a single t″ convolution for the groupedterms, resulting in an efficiency improvement. When thecompute-efficient pairing is determined by the sign of the contributionmade by different ridges to TCC^((r)), the resulting loss in matchingaccuracy is often quite minor.

The set of loxicoherent systems used in the embodiment describedcomputationally by FIG. 20H (these loxicoherent systems being the Lterms of the second summation appearing in this equation) are welladapted to matching the TCC portions that are poorly matched by thecoherent system set that is also employed in this embodiment (the latterbeing represented by the N OCS terms in the first summation appearing inFIG. 20H). Moreover, when determining the T′ and T″ kernels it isparticularly desirable that TCC^((r)) be matched accurately at Δf=0, notonly because the residual error from the N OCS terms in the firstsummation of FIG. 20H is largest there, but also because the DCharmonics associated with this portion of the TCC have a particularlydeleterious effect on the accuracy of lithographic image calculations,as will be discussed. In basic embodiments each loxicoherent system ispreferably constructed to reduce the remaining TCC error as strongly aspossible, and, as will be discussed, it is further desirable that boththe first loxicoherent system (represented computationally byT_(Loxicoherent,1)), and the first rotated system T_(Rotated,1) fromwhich T_(Loxicoherent,1) is derived, be designed to fully eliminateTCC^((r)) at DC intensity harmonics where Δf=0, while in additionminimizing TCC^((r)) throughout the doubled Hopkins domain, subject tothis DC-matching requirement. Loxicoherent systems can meet both goalssimultaneously, since they contain two independently optimizableconstituent lens system apertures, i.e. two independently optimizablekernel functions.

An example illustrating these points is presented in FIG. 21, whichshows a first loxicoherent system that has been extracted from (and is abest approximation to) the TCC^((r)) shown in FIG. 14. Comparison of thetwo figures shows that even the single loxicoherent system depicted inFIG. 21 provides on its own a reasonably close match to the residual TCCleft unmatched by N=24 optimal coherent systems (FIG. 14). Depictionslike FIGS. 14 and 21 can only show TCC^((r)) within a sub-manifold oflimited dimension; in particular, they depict the 2D slice from the full4D TCC^((r)) that governs the imaging of 1D patterns. However, eventhough FIG. 21 can only show this limited 2D slice, the firstloxicoherent system that is partially depicted will approximately matcha large portion of the full 4D TCC^((r)) of which FIG. 14 is a 2D slice;in particular, the first loxicoherent system partially depicted in FIG.21 will be shown below to closely approximate the residual TCC over afull quadrant of the doubly-dimensioned Hopkins domain. It will furtherbe shown that the FIG. 21 loxicoherent system exactly matches the FIG.14 TCC^((r)) at Δf=0, absent small numerical errors. In addition to thisexact matching at the fin peak, comparison of FIG. 21 with FIG. 14illustrates that the single loxicoherent system also provides a closeoverall rendition of TCC^((r)) throughout the doubled Hopkins domain(though in 2D a matching as broad as this may require four loxicoherentsystems instead of one, e.g. one system for each quadrant of the fullHopkins domain). The improvement in image accuracy that results from useof such loxicoherent systems will be discussed and illustrated below. Inbrief, the discussion above in connection with FIG. 1A shows that whenthe novel decomposition employed by the invention is able to accuratelymatch the TCC of a lithographic imaging system, the invention will as aresult be able to accurately match the images formed by the lithographicsystem, allowing the invention to provide accurate dimensionalcompensation to the mask shapes that the lithographic system projects.

Having demonstrated via FIG. 20 and associated discussion that the novelloxicoherent systems of the invention can be applied very rapidly tocalculate image contributions from particular frames of mask data, andhaving shown (in a preliminary way, at this point) that thedecomposition systems of the invention can match lithographic systemswith greater accuracy and efficiency than prior art coherent systems,the question of determining specific loxicoherent systems that suitablymatch a particular lithographic system is next considered. FIG. 22describes steps by which T′ and T″ may be constructed to achieve anessentially optimal TCC^((r)) reduction. As a starting point, FIG. 22Aindicates that T_(Rotated,1) should approximate TCC^((r)) as closely aspossible, and that in a preferred embodiment a single T_(Loxicoherent,1)term should be constructed from T_(Rotated,1) that best retains thisclose approximation. The right-side expression in FIG. 22A writes outthe computational factors in T_(Loxicoherent,1) explicitly, representingthe physical structure of this system as a constituent coherent system(represented by T′ twice repeated) that is paired with a constituentincoherent system (represented by T″). Prior to extraction of the firstT_(Rotated) term, the residual TCC will be strongly concentrated inclose vicinity to the Δf=0 “fin ridge”, and may be considered to havenear-negligible value elsewhere. This means that when T_(Rotated,1) ismatched to TCC^((r)), {tilde over (T)}(Δf) will typically be close tozero at frequency pairs whose distance from the fin is appreciable incomparison with the structural scale in the illumination coherencepattern, or the lens resolution. The equation of FIG. 22A expresses thisbehavior quantitatively, defining the lens resolution as beingcomparable to the ratio of the numerical aperture (NA) to thewavelength, and then defining the associated range of relevant(fin-impacted) frequencies (in inverse-distance units) as the reciprocalof this quantity. Since {tilde over (T)} and T_(Rotated,1) fall to zeroat large Δf in matching fashion, T_(Rotated,1) will provide a goodoverall approximation to TCC^((r)) if

⁽¹⁾ is determined from TCC^((r)) in the vicinity of the fin. The valueattained by

at larger distances from the fin will not significantly impact the fitquality, because at such points a properly set {tilde over (T)} willensure that T_(Rotated) takes on a suitably low value to matchTCC^((r)), making the fit insensitive to the value of

.

Beyond these general considerations, it is highly desirable in fastcalculations of lithographic images that any approximate rendering ofTCC^((r)) provide a particularly close matching at Δf=0, since, as willbe discussed, image accuracy is usually quite sensitive to the accuracywith which this portion of the TCC is rendered.

FIGS. 22C, 22D, and 22E explain properties of TCC^((r)) and

⁽¹⁾ that aid this matching. In particular, the first line of FIG. 22Cconsiders the FIG. 1E Mercer series in the theoretical limit where aninfinite number of kernels is used, deriving from it in the second linean exact series expression for the residual TCC after N OCS systems havebeen extracted. As discussed, we can arbitrarily set the scale of {tildeover (T)}⁽¹⁾ by choosing {tilde over (T)}⁽¹⁾(0)=1, and from FIG. 16B wesee that T_(Rotated,1) will exactly fit TCC^((r)) at Δf=0 if we then set

equal to TCC^((r)) along the “ridge of the fin”, i.e. set

to match TCC^((r)) at f₁=f₂=f. FIG. 22D shows how such an assignment canbe related to the frequency-domain OCS kernels by substituting from theequation of FIG. 22C. In particular, FIG. 22D shows that

⁽¹⁾ will be equal to the sum of the squares of those OCS kernels thatconstitute the residual TCC (these excluded kernels having beendiscarded when the employed set of N coherent systems was chosen). As asum of squares,

⁽¹⁾(f) will be a real-valued and non-negative quantity, a propertyexpressed algebraically in FIG. 22E. FIG. 22E also notes that TCC^((r))will likewise be real-valued and non-negative along the ridge of thefin.

The non-negativity property of the

kernel will next be used in the specific numerical determination of

as a function of f. It has been shown in connection with FIG. 16B that,for the first rotated kernel,

at a particular mean-frequency f≡(f₁+f₂)/2 should be set equal toTCC^((r))(f,f), i.e. to the residual TCC value along the fin ridge atthis same mean frequency. Also, FIG. 22B indicates that {tilde over (T)}will typically suppress whatever contribution

happens to make, except in regions where f₁≅f₂, which means that thevalues taken on by

are, for the most part, only relevant when f₁≅f₂ (though f₁ and f₂ neednot be exactly equal). This justifies a general application of theresult shown in the first line of FIG. 22F, which points out that whenf₁ and f₂ are reasonably close to each other, the TCC^((r)) value alongthe fin ridge at the arithmetic mean of these two frequencies will bevery close numerically to the geometric mean of the two values taken onby TCC^((r)) at these two nearby frequency locations along the ridge. Atother locations where f₁ and f₂ are strongly different we expect therotated system to suppress any significant contribution, since {tildeover (T)} will be very small at such locations (per FIG. 22B).Conversely, at frequency pairs where {tilde over (T)} allows

to make a significant contribution, FIG. 22F points out that the valueof TCC^((r)) when both its arguments are set to the (arithmetic) meanfrequency (f₁+f₂)/2 will be very close to the geometric mean of the twoTCC^((r)) values obtained by setting both arguments first to f₁, andthen to f₂. This follows because TCC^((r)) exhibits only finitecurvature along f meridians, such as along the ridge of the fin, eventhough its curvature along the orthogonal Δf meridians is essentiallyinfinite at Δf=0 (per FIGS. 5B and 5C). As a result, the value thatTCC^((r)) takes on at the ridge location where both its arguments areequal to the arithmetic mean of f₁ and f₂ will be approximately equal tothe geometric mean of the TCC^((r)) ridge values at f₁ and f₂, assumingthat the {f₁, f₂} frequency pair is one at which T(Δf) is significant,i.e. that f₁≅f₂. (Only frequency pairs at which T(Δf) is significant areimportant in setting

(f), i.e. we need to match TCC^((r)) in the vicinity of the fin.) Notethat although the TCC^((r)) factors arising in the first line of FIG.22F are being evaluated along the fin ridge, i.e. with the samefrequency being used for both the first and second argument of theTCC^((r)) function, the two frequencies f₁ and f₂ under considerationare generally not exactly equal, though FIG. 22B shows that these twofrequencies will not be greatly different when the numericalcontribution of T_(Rotated,1) has significant magnitude. In other words,the frequency pairs of interest for determining

are those lying within the fin or its vicinity, since T can be relied onto block contribution by

at large distances from the fin, thereby achieving an appropriatelysmall value for T_(Rotated) away from the fin where TCC^((r)) islikewise small; for this reason {tilde over (T)} makes the value takenon by

unimportant away from the fin. Nonetheless, it should be understood thatthe relevant frequencies need not reside exactly on the fin peak. ThoughTCC^((r))[f₁, f₂] will generally exhibit rapid variations in the finregion of the Hopkins domain, the variation along the arc connecting theparticular frequency pairs used in the first line of FIG. 22F will bemore gradual, making the approximation in the first line of FIG. 22Fvery accurate. Moreover, this approximation will be exact for the mostcritical frequency pairs, namely those which do lie along the fin ridge,where f₁=f₂. This latter point means that even though FIG. 22F involvesan approximation, the loxicoherent system generated from FIG. 22Fachieves a near-optimal extraction of TCC^((r)), as will be shown.

The second line of FIG. 22F then substitutes from the first part of FIG.22D, and the third line makes the further modification of replacingTCC^((r))(f₂, f₂) by its complex conjugate, which is a validsubstitution according to FIG. 22E.

The next step in a preferred approach for obtaining the firstloxicoherent system is to use FIG. 22F to express T_(Rotated,1) in aform exhibiting the general loxicoherent structure shown in FIG. 20B,which essentially means decomposing

⁽¹⁾(f) into suitable separated T′ functions of f₁ and f₂. The first twolines of FIG. 22G express this decomposition, including substitutionfrom the last line of FIG. 22F. Note that the last line of FIG. 22Falready achieves a separation of

. This means that the first approximation in the second line of FIG. 22G(based on substitution from FIG. 22F) already provides a structure thatqualifies as a loxicoherent system; however the right side of the secondline indicates that because this interim system is not yet fullydeveloped, T″ should only be regarded as being implicitly determinedpending further refinement, meaning more specifically that in apreferred embodiment T″ will be chosen in such a way as to makeT_(Loxicoherent,1) as accurate a rendition of TCC^((r)) as possible, andindicating more generally that the optimum T″ will therefore bedifferent from the optimal {tilde over (T)}, as will be discussed.

If on this basis one identifies T′ with the square root of TCC^((r))along the peak of the fin, as expressed in the third line of FIG. 22G,the resulting loxicoherent system will exactly match TCC^((r)) along thecritical Δf=0 fin peak, assuming the normalization choice T″(0)=1. Thethird line of FIG. 22G thus represents a preferred method fordetermining the first T′ mask filter, thus defining the constituentcoherent system of the first or primary loxicoherent system(T_(Loxicoherent,1)) that is employed by the invention. In brief, theequation of FIG. 22G decomposes the T_(Rotated) kernel which lies alongthe rotated axis (f₁−f₂)/2 into a separated bilinear product of the T′mask filters, namely T′(f₁) T′(f₂).

To complete the determination of T_(Loxicoherent, 1), we shouldpreferably choose a T″ kernel that optimally takes into account thechange made by replacing

in the rotated system precursor with a separated pair of T′ kernels.FIG. 22G guarantees the key property that T_(Loxicoherent, 1) matchTCC^((r)) on the ridge of the fin, as long as T″(0) is set to 1. T″ atother values of Δf can then be chosen in such a way as to optimize thematching of T_(Loxicoherent, 1) to the remaining regions of TCC^((r))(i.e. the regions where Δf≠0).

FIG. 22H provides a least-squares metric E_(Filter) to accomplish thismatching, with the matching criterion being the minimization ofE_(Filter) with respect to differential variations in the function T″(f)at the optimum. In particular, E_(Filter) at the optimum (i.e., atminimum matching error) should exhibit no first order change when smallvariations are introduced in T″, with these variations having the formshown in FIG. 22I. The δ-function perturbation in FIG. 22F is introducedat an arbitrary location Δf_(pert), i.e. at Δf=Δf_(pert).

Another consideration here is that the loxicoherent system is inherentlyunable to match any TCC^((r)) value with non-zero magnitude that mayhappen to arise at a frequency pair where either one of the T′ factorsin FIG. 22H is zero. Thus, if there is a Δf value with the property thatT′(f₁) and/or T′(f₂) is zero for every {f₁, f₂} frequency pair havingthis Δf difference, then E_(Filter) will automatically be stationary,and T″ should preferably be set to zero at this Δf value. To identifysuch conditions a windowing function D(f) has been introduced in FIG.22H. In some embodiments, D(f) is defined to be 1 whenever T′(f) hasappreciable magnitude, and 0 whenever T′(f) is 0. Alternatively, D(f)may be set to 0 whenever the magnitude of T′(f) falls below a designatedsmall threshold. Other useful designs for window D(f) will be discussedin the context of matching the 4D TCC that governs 2D patterns.

In the special case where the D(f₁)D(f₂) product in FIG. 22H iseverywhere zero for some particular value of f₁−f₂, T″(f₁−f₂) can be setto zero by definition. Ordinarily this special circumstance does notapply, and it then becomes straightforward to invert the condition ofstationarity in E_(Filter) to solve for T″, using steps that are broadlysimilar to those discussed in connection with FIG. 12. The resultingsolution is shown in FIG. 22J.

The equation of FIG. 22J has been generalized to include a parameter p,whose value can be set to 1 if strict minimization of E_(Filter) issought. However, parameter p can also be set to a lower value, such as0, if one seeks a T″(Δf) solution with reduced content at large |Δf|.Although the p=1 direct solution to the minimization will yield anaccurate calculation of the image intensity via FIG. 20H, it is possiblein principle for the equation of FIG. 20H to yield small negativeintensity values in very dark areas of the image. While such values donot represent a large error in numerical terms (since the loxicoherentsystem in fact acts to reduce the absolute error), and are quite rare,it may be considered preferable to ensure that all intensities arenon-negative. A simple way to achieve this is to threshold I(x) to benowhere below zero. However the possibility of small negativeintensities can be rendered more remote by lowering parameter p. Settingp to zero corresponds to determining T″(Δf) by matching the averages ofTCC^((r)) and the T′ product along each contour of constant Δf. With anychoice of p between 0 and 1, the loxicoherent kernel T₁″(Δf) for thefirst loxicoherent system will typically have a similar shape to {tildeover (T)}₁(Δf), and will be strongly peaked at Δf=0.

When applying FIG. 22H to the first loxicoherent kernel it may not benecessary to include the D factors (i.e., the D factors may simply beconsidered to have value 1 throughout the bandwidth), but it should benoted that FIG. 22J is applicable to other loxicoherent kernels beyondthe first. For example, it will be shown that the D factors can beuseful when calculating multiple sets of loxicoherent systems for 2Dspatial frequency pairs (i.e., for the 4D TCC).

It should also be noted that even though the method of FIG. 22 sets T′in order to provide a good match to TCC^((r)) in the vicinity of thefin, FIG. 22J optimizes T″ to provide a good match to TCC^((r))throughout the doubled Hopkins domain. While a prior art coherent kernelmight be considered to achieve an analogously optimal match over thedoubled domain, in the sense that the TCC eigenfunctions Ψ are theoptimal aperture choice for overall RMS matching of the TCC usingcoherent systems, the prior art coherent kernels are unable toaccurately match the crease region of the TCC. In contrast, loxicoherentsystems in accordance with embodiments of this invention have a richerstructure comprising at least two distinct kernel functions, and thisricher structure allows them to provide both advantages simultaneously.

FIG. 23 depicts two plots showing the loxicoherent kernel t″ and maskfilter T′ obtained using the FIG. 22 method, for the C-quad test casediscussed previously, which employs the source shown in FIG. 9B. The T″kernel was calculated with p set to 1 in FIG. 22J. The inputs to FIGS.22G and 22J were obtained from the TCC^((r)) kernel shown in FIG. 14,which was also used to obtain the T_(Rotated) system shown in FIG. 13.(In addition, the FIG. 23 kernels are used in example image simulationsdiscussed below.) FIG. 23 shows t″ in the spatial domain, and the plotillustrates that loxicoherent t″ kernels have a characteristic scalethat is considerably broader than the lens resolution, often being onlyslightly smaller than the size of the optical ambit, which in thisnon-limiting example is 2 microns. The mask filter T′ exhibits acomplicated dependence on spatial frequency, determined primarily by thesource shape, as will be discussed.

Application of a loxicoherent system (e.g., calculating and applying thel=1 term of the second sum in FIG. 20H) might be considered to bear someanalogy with a generic calculation of incoherent optical flare, in thelimited operational sense that both procedures involve the convolutionof a kernel having broad spatial extent with a quadratic function of themask pattern (though t″ for a loxicoherent system is generally much lessextended than typical optical flare kernels, and in addition the resultof the squared t′ convolution has little resemblance to the lithographicimage intensity which drives flare, since the circular aperture of thelithographic lens is very different from the T′ aperture of theconstituent coherent optical system, and because the lithographic imageis produced with a partially coherent source). Since the firstloxicoherent system maps TCC^((r)) quite closely, it could be said thatOCS truncation error is somewhat flare-like in its behavior. However, itshould be noted that the OCS truncation error has only a broadresemblance to flare or dose error. First, the t′(x) pre-filter has astrong frequency dependence which differs very substantially from thatof OCS kernels, and thus the intensity pattern produced by the optimalconstituent coherent system is very different from the lithographicintensity which drives optical flare, e.g. via scatter or strayreflections. (In other words, though the constituent coherent system maybe optimal for its role as one element of a best-matching loxicoherentsystem, the image produced by this constituent coherent system will, ifconsidered in isolation, generally bear little resemblance to either theloxicoherent image contribution, or the lithographic image.) Second, thebroadly flare-like t″(x) loxicoherent kernel has a fine structure withno analog in flare (as may be seen in the left plot of FIG. 23), and hasa fall-off width that is usually somewhat shorter than the opticaldiameter, whereas the kernel for optical flare usually has a much largerextent. And, of course, the procedure for obtaining the loxicoherentsystem kernels has no resemblance to procedures for determining flarekernels, since the physical quantities and mechanisms involved arecompletely different. Moreover, the loxicoherent systems of the presentinvention encompass a broader range of structures than the FIG. 22preferred embodiment for the first loxicoherent system, and thesealternative systems no longer take the form of a convolution with asquared mask transmission analogue, as will be discussed. For example,the DC-monolinear system discussed below may be regarded as being evenmore closely analogous to a truly coherent amplitude image than is anOCS/Mercer system, since the latter is a fully quadratic function of themask amplitude transmission, whereas the DC-monolinear system maysometimes be considered to exhibit a quasi-linear dependence on m(x)(see below). Application of the DC-monolinear system thus lacks even anoperational resemblance to a calculation of incoherent optical flare.Nonetheless, the first loxicoherent system provides the most importantimprovement over a truncated OCS decomposition, and its rough impact inthe spatial domain is to correct a pattern-dependent dose-like error.

FIG. 24 depicts the remaining residual TCC after the first loxicoherentsystem (whose kernels are shown in FIG. 23) has been extracted from theresidual TCC of FIG. 14, in accordance with the invention. Since FIG. 14depicts the residual TCC after 24 standard OCS systems have beenextracted from the exact TCC, one may regard FIG. 14 as depicting thefrequency-domain TCC error that is imposed when the standard FIG. 1D OCSdecomposition of the prior art is applied (with N=24 OCS systems beingused in this case). FIG. 24 then depicts an improved residual TCC errorthat results from applying FIG. 20H with N=24 and L=1. It is apparentfrom a comparison of FIGS. 24 and 14 that the introduction of a singleloxicoherent system to the TCC decomposition has resulted in a verysubstantial reduction in residual error.

It should be noted that the adoption of each additional loxicoherentsystem in FIG. 20H will entail a computational cost of two new FFT-basedconvolutions, whereas the addition of each single conventional OCSsystem will only require a single such convolution. However, when N isof typical magnitude, the accuracy gain from the first few loxicoherentsystems (and in particular from the first loxicoherent system) is verylarge, since these systems are tailored to extract those portions ofTCC^((r)) that are most recalcitrant to standard OCS decomposition, withthe image error associated with these remaining TCC^((r)) portions thusbeing difficult to mitigate by increasing N. For this reason theimprovement provided by the first loxicoherent systems comes atconsiderably lower computational cost than would be entailed by adoptionof sufficient coherent systems to achieve the same accuracy. In otherwords, the loxicoherent systems allow a given accuracy target to beachieved at an appreciably lower computational cost.

To illustrate this advantage, FIG. 25 depicts the residual TCC from anOCS decomposition for the same imaging configuration as the OCS residualof FIG. 14, except that in FIG. 25 two additional OCS kernels have beenused, i.e., 26 standard OCS kernels have been used in obtaining FIG. 25,whereas FIG. 14 uses 24. The computational cost of obtaining the FIG. 25residual is thus the same as that for the FIG. 24 loxicoherent residual,since the computational cost of each added loxicoherent system is twoFFT-based convolutions. However, the two added OCS convolutions in FIG.25 are seen to provide only a modest decrease in the TCC error comparedto FIG. 14, reflective of the fact that in practical applications theconventional OCS expansion is operating in a regime of diminishingreturns. The FIG. 24 residual TCC likewise entails an added cost of twoFFT convolutions relative to FIG. 14, but in the case of FIG. 24 the twoconvolutions are used to incorporate a loxicoherent system into thedecomposition, in accordance with the invention. FIG. 24 shows that theloxicoherent system provides a far greater accuracy improvement than doadditional standard coherent systems having the same overall cost.

The dashed line along the Δf=0 crease in FIG. 24 is entirely flat, at aheight of 0, illustrating that the residual TCC error at Δf=0 iscompletely eliminated by the introduction of a loxicoherent systemconstructed according to FIGS. 20B, 22G, and 22J. The benefit fromachieving such exact correction along the ridge peak of the fin may beunderstood by reference to the equation in FIG. 26, which can readily bederived by applying Parseval's theorem to the equation in FIG. 20G. Theleft side of FIG. 26 is the total squared error in an image whencalculated using a truncated OCS expansion, while the right side is anestimate of that total error based on the approximation that theresidual TCC which gives rise to the total error can be approximated bythe first loxicoherent system.

An important consideration in assessing the impact of FIG. 26 is that inpractice the filtered mask autocorrelation (the expression in squarebrackets) will almost always be strongly peaked at Δf=0. Several aspectsof current lithographic practice contribute to this dominance of thezero frequency in the autocorrelation. For example, when thespatial-domain circuit pattern contains semi-isolated small features, itis known that these features should be surrounded within the designedmask pattern m(x) by a quasi-periodic array of even smaller non-printingfeatures, of the kind known in the art as assist features or SRAFs (forSub-Resolution Assist Features). SRAFs provide an extension in depth offocus, and for this reason their use in modern lithography has becomequite standard. Though in many cases these SRAFS are kept too narrow toprint as resolved patterns, they are nonetheless able to concentrate themask spectral content M(f) into spatial frequencies that have largedepth of focus, while depleting spatial frequencies with small depth offocus. This is one of several reasons why mask content that is printedusing state-of-the-art lithography tends to favor preferred pitches, andto be deficient in so-called forbidden pitches. As will be discussed,such behavior increases the sharpness of the Δf=0 peak in the maskautocorrelation term of FIG. 26. In addition, the decreased use inrecent years of strong mask phase shift, and the increased use ofbright-background masks (particularly in so-called negative-toneprocesses), both tend to produce a large disparity between the magnitudeof the zero (i.e. DC) order and all other orders. This is illustrated inFIG. 27, which shows the distribution of energy within the 2D orders of1000 example mask clips from an integrated circuit layer referred to inthe art as a 22 nm first metal layer. FIG. 27A plots this frequencydomain energy on a logarithmic scale, where it is seen that the zeroorder (in x and y) predominates over all others. The height of the {0,0}peak in FIG. 27A is about 0.234, while the next strongest orders onlyhave intensity 0.006. While the strong zero-order is often somewhatdiminished when the mask filter T′(f) is applied to M(f) to produce thefiltered spectrum M′(f), the T′ filter will typically impose a complexstructure of its own on the filtered spectrum M′, which furtherincreases the autocorrelation peak at the origin. In addition, with 2DIC patterns the diffracted energy is usually highly concentrated alongpreferred directions, which are most often directions that are tiltedalong the x or y axes, corresponding to the main diffracting meridiansof so-called Manhattan geometries. This tendency can be seen in FIG.27B, which plots the same data as FIG. 27A on a linear scale, but withthe {0,0} order removed. It is seen that orders which do not lie withinthe meridians of the x or y axis are relatively attenuated.

All of these factors tend to provide the filtered spectrum profile M′(f)with a highly non-uniform structure. This in turn means that the twoautocorrelated M′ terms appearing in FIG. 26 will make their strongestcontribution to the autocorrelation when they are “aligned”, i.e. whenΔf=0. The T″(Δf) kernel is also very sharply peaked at Δf=0; forexample, T″ will typically resemble the {tilde over (T)}(Δf) functionshown in FIG. 13B, as will be discussed. FIG. 26 then shows that the RMSintensity error as averaged over the optical diameter will tend to bevery strongly dominated by the behavior at Δf=0, making it desirablethat the first loxicoherent system exactly match TCC^((r)) at the peakof the fin, as is accomplished when FIG. 22G is used. Simulations showthat image accuracy will actually be slightly diminished when bothkernels in T_(Loxicoherent,1) are instead optimized to best matchTCC^((r)) in an averaged way across the bandpass, as opposed to choosingthe FIG. 22G match at Δf=0. Of course, the fin will typically constitutea very substantial portion of the overall TCC error, making thenumerical difference between these two matching criteria quite small.

FIG. 28 is a Table showing the 1D accuracy of the improved method ofthis invention versus the conventional approach. The Table shows RMS andworst-case intensity errors over a broad spectrum of CDs. The imagingconditions are those of the C-quad test case discussed above, e.g., inconnection with FIG. 9. Results for two categories of 1D patterns areshown, namely 1) so-called equal line/space patterns, where the dutycycle of the periodic pattern is kept at 50%, with the pitch beingstepped from 70 nm to 1000 nm in increments of 2.5 nm, and 2) a set ofso-called isolated space patterns, each being a single isolated opening,with the widths of the isolated open features being stepped from 35 nmto 500 nm in increments of 1.25 nm. The table rows list various metricsthat describe the error when calculating the intensity over extendedcutlines. The right-most column shows these errors when the image iscalculated according to the invention, using FIG. 20H with N set to 24and L set to 1, i.e. by extracting a first loxicoherent system from theTCC residual left by 24 OCS systems (or kernels). Columns further leftin the table show the error levels achieved by standard OCS usingdifferent numbers of kernels. The rightmost of these OCS columns (i.e.the second column in from the right) shows the error levels achieved byemploying 80 OCS kernels, and in addition taking the step ofartificially anchoring the dose at a level which minimizes the intensityerror (instead of following the standard practice of adjusting the doseto print a critical feature on target).

Overall, the FIG. 28 Table shows that the loxicoherent systems of thepresent invention provide broadly superior accuracy to prior art OCS, atsignificantly lower compute cost. If the N=24 OCS system is taken to berepresentative of current practice, column 2 of the Table shows that aworst-case intensity error of about 1% is incurred. In contrast, column6 shows that use of the invention allows a stringent 0.25% accuracycriterion to be met (vis-à-vis count-truncation error) with littleadditional overhead. Alternatively, the improved method can be used tomaintain currently accepted (and anticipated future) accuracy levelswith far fewer FFTs being required, and thus with correspondingly fastercalculation times.

The FIG. 28 results illustrate the very dramatic reduction in residualimage error that is achieved by extracting only a single loxicoherentsystem, i.e., by merely choosing L=1 in FIG. 20H. It will prove usefulto understand the mechanism behind this dramatic improvement in moredetail. The single loxicoherent system is seen not only to remove nearlyall of the residual TCC error, but also to reduce TCC^((r)) moreeffectively than is possible with quite a substantial number of addedconventional coherent kernels. FIG. 24 illustrates that some residualTCC error does remain, but it will be shown that, within a certainasymptotic level of approximation, the single loxicoherent system isable to extract (in the idealized asymptotic limit) the entirety ofTCC^((r)), as only an arbitrarily large number of conventional coherentsystems could similarly extract. Such a complete level of success isonly achieved as an approximation, and the single loxicoherent systemdoes in fact leave some residual TCC error. While this residual error isquite small, it can be further reduced by extracting additionalloxicoherent systems, and to design these additional systems it isuseful to exploit the mechanism by which the first loxicoherent systemis able (in the idealized asymptotic limit) to match the performance ofan extremely large number of coherent systems.

In particular, to develop efficient loxicoherent systems beyond thefirst, it is useful to exploit the same physical considerations whichallow the loxicoherent system of FIG. 20B to achieve almost-completeextraction of TCC^((r)), dramatically outperforming the much slowerextraction achieved by additional coherent systems of the prior art FIG.1B form. These physical considerations may be understood with referenceto FIG. 29, wherein FIG. 29A encapsulates the extreme asymptoticbehavior of TCC^((r)) in the regime of diminishing returns that arisesafter a great many conventional coherent kernels have been extracted. Asdiscussed, TCC^((r)) takes on the character of a very narrow fin in thisregime, and because of the narrowness of the fin we may consider it tohave almost a δ-function width; thus FIG. 29A represents the (not fullyrealizable) limit where such ideal asymptotic behavior is considered toactually be realized.

In this limit, an asymptotic form for the OCS kernels may readily bededuced by using the equation of FIG. 29B, whose rationale will beexplained. The first line of FIG. 29B considers the integral ofTCC^((r)) with a certain delta-function whose key role (explained below)is emphasized by enclosing it in braces; the right side of the firstline of FIG. 28 shows the result of approximating TCC^((r)) in theasymptotic limit by substituting from FIG. 28A. The second line of FIG.28B then notes that the resulting integral can be carried outexplicitly, with the result being the right-hand expression in the 2ndline. It can then be observed that the left-most and right-most terms ofthe second line of FIG. 29B have the form of an eigenfunction equationfor the TCC^((r)) operator, with the function in braces serving as theeigenfunction. Thus, this asymptotic eigenfunction consists of aδ-function centered at a frequency f_(j). This means that if theasymptotic limit of the equation of FIG. 29A could actually be reached,the OCS kernels would take the form of δ-functions in the frequencydomain. If we then consider a situation where TCC^((r)) is the residualTCC after j−1 OCS kernels have been extracted, and presume that thesecoherent kernels have been optimal, i.e. eigenfunctions of the TCC, thenthe jth OCS kernel will by definition correspond to the eigenvalue thatis dominant in the Mercer series for the remaining TCC^((r)), andbecause this eigenfunction (in the asymptotic limit) is a δ-functioncentered at f_(j), we see that f_(j) must be positioned at the highestremaining point in the fin structure of TCC^((r)) (since the eigenvalue

(f_(j)) is maximal there), as is noted in the fourth line of FIG. 29B.

Thus, in the asymptotic limit, an OCS kernel only succeeds in extractingthe residual TCC at a single frequency f_(j) along the fin, leaving theerror at all other frequencies along the fin unextracted. Since theresidual TCC is highly extended along the diagonal, it is seen that thejth OCS kernel is quite inefficient at reducing the overall remainingTCC error, despite f_(j) being the largest single point of residual TCC.This helps explain why the prior art OCS method faces diminishingreturns once N becomes large, and will also prove useful in designinghigher-order loxicoherent systems to achieve even greater levels ofimage accuracy than those illustrated by FIG. 28. Of course, it isunderstood that FIG. 29A only represents the limiting asymptoticbehavior of TCC^((r)), and this limiting behavior cannot actually bereached at finite j. For example, the OCS kernels must be smoothfunctions as noted above, and even though frequency-domain versions ofthese kernels tend to exhibit comparatively sharp localizations withinthe pupil at large j, they cannot truly take the form of δ-functions.Moreover, a more careful version of the FIG. 29 analysis shows that thesimple form of FIG. 29B does not hold when more than one point along thefin has the same magnitude (though the same overall conclusion isreached in the end), and points of matched fin height occur very oftenin practice, given system symmetries. Nonetheless, FIG. 29B doesappropriately reflect both the strong pupil localizations (which usuallyare multi-fold within each kernel) and the slow extraction rate that areexhibited by high-order OCS kernels.

It should also be noted that even though each Mercer term formed fromsingle OCS kernels in the FIG. 29B limit only succeeds in extracting avery small portion of the TCC^((r)) fin, such kernels by definitionrepresent the optimal coherent match to TCC^((r)), and a major part ofthis matching success arises in the extended regions of the doubledHopkins domain that are not part of the fin, where the very lowTCC^((r)) levels that have already been achieved must not be undone whennew systems that are added. In particular, it will now be shown that thedelta-function asymptotic OCS kernels of FIG. 29B accurately reproducethe near-zero-valuedness that TCC^((r)) exhibits away from the fin inthe asymptotic regime. This point is demonstrated in FIG. 29C, where thefirst line reiterates that TCC^((r))(f₁,f₂) is approximately zero inregions where f₁≠f₂, once a large number of OCS kernels have beenextracted. And, per the second line of FIG. 29C, this behavior ismatched by Mercer terms that are formed from OCS kernels of theasymptotic FIG. 29B form, since at such (f₁, f₂) frequency pairs awayfrom the fin one or the other δ-function kernel appearing in the Mercerterm will be zero. The fin region is recalcitrant to OCS extraction, butOCS largely succeeds in reproducing other parts of the TCC once Nreaches moderate values, and this success is maintained as new OCSsystems are added, i.e. succeeding OCS systems continue to holdTCC^((r)) near zero away from the fin.

This matching success away from the fin would be lost if one attemptedto extract more than one position along the fin at a time using only asingle putative OCS kernel, as is demonstrated in FIG. 29D. Inparticular, FIG. 29D considers the behavior of a trial coherent kernelfunction

that attempts (in contradiction to the equation of FIG. 29B) tosimultaneously extract the fin content at two fin frequencies f_(j′) andf_(j″). A Mercer-like term formed from such functions will unfortunatelyfail to match the zero value of TCC^((r)) at the non-fin frequency pair(f_(j′), f_(j″)), as shown in the second line of FIG. 29D. If we assumefor simplicity that the fin heights at f_(j′) and f_(j″) are not thesame, such a term cannot match TCC^((r)) as accurately as the FIG. 29Bvalid eigenfunctions. This demonstrates that even an optimal coherentsystem is inherently limited to a very slow extraction rate in the largekernel-count regime, and these idealized asymptotic arguments prove tobe qualitatively accurate as limiting trends in practical regimes.

The situation with loxicoherent systems is quite different, as explainedin FIG. 29E. The second line of FIG. 29E shows that each T′ factorappearing in a loxicoherent system can be interpreted as a superpositionof a very large number of δ-function samples. This means that an optimalT′ for a first loxicoherent system that is constructed per FIG. 22G canbe regarded as a superposition of δ-functions which sample every pointalong the fin (unlike an optimal coherent system, which only samples asingle fin point in this asymptotic limit). However, as shown in thethird line of FIG. 29E, such a superposition will not lead to a poormatching of T_(Loxicoherent) to TCC^((r)) away from the fin (wheref₁≠f₂), as arose with the putatively multi-point coherent system of FIG.29D (which therefore proved suboptimal), because loxicoherent systemsinclude a T″(f₁−f₂) term which acts as an “envelope” that (at least inthe asymptotic limit) drives the loxicoherent system to zero atfrequency pairs away from the fin. Along the fin, i.e. when f₁=f₂=f, ithas already been shown that the loxicoherent system will matchTCC^((r))(f,f) exactly when constructed according to FIG. 22G, as notedin the fourth line of FIG. 29E. FIG. 29E thus shows that in theasymptotic limit the loxicoherent system will essentially matchTCC^((r)) at all {f₁,f₂} frequency pairs, i.e. the match will(asymptotically) be perfect throughout the doubled domain.

Loosely speaking, FIGS. 29A-29E demonstrate that even though a coherentsystem which is fully optimal can only extract “a single point” of theTCC^((r)) fin, the richer structure of the loxicoherent system allows itto extract “all points at once”. In particular, the constituent coherentsystem in the loxicoherent system is able to extract all points alongthe fin at once, in contrast to the prior art OCS/Mercer coherent systemwhich (in the asymptotic limit) can only reduce TCC^((r)) at a singlepoint; this strongly superior performance is achieved because theconstituent coherent system acts in sequence with the constituentincoherent system, with the latter preventing the degradation of fittingaccuracy that arises at f₁≠f₂ when an OCS/Mercer coherent systemattempts to extract more than a single fin point. It should bereiterated that FIG. 29A only describes an asymptotic limit that is notmet in realistic cases, and that the behaviors derived from FIG. 29A areonly qualitatively accurate. While the first loxicoherent system willtypically achieve a dramatically larger extraction than can additionalcoherent systems, as illustrated in FIGS. 24 and 28, these examples alsoshow that the first loxicoherent system cannot generally be expected totruly extract the entire residual TCC, as nominally occurs in theasymptotic limit. However, the first loxicoherent system does achievevery substantial reductions in TCC^((r)) by means of the FIG. 29Emechanism, and this same mechanism can be exploited in designingadditional loxicoherent systems that achieve further significantreductions in TCC^((r)), as will be discussed.

A further qualitative guideline involving the appropriate choice of Nmay be inferred from the difference in asymptotic behavior of eachadditional coherent system that is introduced if N is further increased,as compared with the asymptotic behavior of the first loxicoherentsystem. Once N has reached the regime where additions to the coherentsystem set will roughly follow the FIG. 29C form, each added coherentsystem merely provides an incremental reduction in TCC^((r)), throughextraction of only a single (though largest) remaining point along thefin, whereas per FIG. 29E the asymptotic behavior of the lowest orderloxicoherent system is to extract all fin points at once, thusaccomplishing the same result as would inclusion of all successiveMercer terms, i.e. N→∞. In particular, we may infer from this asymptoticbehavior that, were it to be followed rigorously instead of onlyqualitatively, there would eventually be no benefit from furtherincreasing the number N of coherent terms that, per FIG. 20H, areextracted as a precursor step to calculating the first loxicoherentsystem. This conclusion holds asymptotically because the singleloxicoherent system eventually becomes able to provide the same benefitas would any indefinitely large number of added coherent systems, i.e.the first loxicoherent system removes the entirety of the fin,regardless of the specific value of N (once N is very large). Thoughthis behavior will only obtain qualitatively in practice, one may findin a typical case that after N reaches a value of about 50 or 100, thereis little merit in further increasing N before extracting the firstloxicoherent kernel. This means that further increases in N willtypically have passed into a regime of diminishing returns at thatpoint, even when augmented by a loxicoherent system. Nonetheless, itstill proves possible to obtain additional rapid improvements inaccuracy by extracting additional loxicoherent systems (i.e. byincreasing L rather than N in FIGS. 20H and 20I), as will be discussed.

A comparison of the asymptotic form taken on by the constituent coherentT′ mask filters

$\left\lbrack {{{namely}\mspace{14mu}{T^{\prime}(f)}} = \sqrt{\overset{︵}{T}(f)}} \right\rbrack$and the asymptotic form of optimal OCS/Mercer kernels Ψ_(j) [namely

$\left. {{\Psi_{j}(f)} = {\sqrt{\overset{︵}{T}\left( f_{j} \right)}{\delta\left( {f - f_{j}} \right)}}} \right\rbrack$shows that the constituent coherent system of the first loxicoherentsystem will differ very substantially from a standard OCS coherentsystem, and the same conclusion holds with higher order loxicoherentsystems. This means that if the constituent coherent system were used inisolation, i.e. as a standalone OCS system, its performance wouldgenerally be very poor compared to an optimal OCS system. (While theoptimal OCS system yields only a small asymptotic increase in accuracy,the constituent coherent system would significantly degrade accuracy ifused in isolation.) Similarly, the constituent incoherent system wouldmake poor intensity predictions if used independently from theconstituent coherent system, since the former fails to capture thestrong frequency dependence that is present along the fin (as in FIG.23). However, extremely good performance is obtained by the sequentialpairing of the two constituent systems to form a loxicoherent system.

FIG. 30, consisting of FIGS. 30A, 30B and 30C, is a logic flow diagramillustrating the basic steps with which the invention employs one ormore loxicoherent systems in accordance with the embodiments of thisinvention. The method is depicted in the non-limiting context of an OPCimplementation. FIG. 30 shows that the embodiments of this invention canuse more than one loxicoherent system, and also emphasizes the use ofthe embodiments of this invention as a tool for producing masks usefulfor, by example, the manufacturing of semiconductor integratedcircuits/chips.

FIG. 30A diagrams the functions executed by the invention, in a highlevel summarized form. Some steps in FIG. 30A could be viewed as beingsimilar to those of prior art OPC systems. Block 1100 is a chip-levelsetup procedure, largely known in the art, where options are defined,per user inputs, for specifying target dimensions and edge positions forprinted integrated circuit features, for example, defining these targetdimensions to be those of the patterns explicitly supplied in an inputqueue or database, or specifying that these target dimensions bemodified away from the queued input dimensions according to user-definedrules; also specifying partially coherent imaging conditions for alithographic system; identifying an anchoring mask feature; and definingso-called fragmentation rules. In some embodiments the inventionprepares a queue by extracting designs for circuit chips, or modules, orthe small circuit sections known in the art as “clips”, from a databaseor library, selecting specific library entries based on user input.Block 1100 may be carried out by a gateway node in a large-scalecomputer cluster.

The process of identifying the anchoring mask feature is a well-knownstep in IC mask design. The specific choice of anchor feature isbasically a matter of engineering judgment, but typically one chooses asimple yet key pattern whose preferred mask design can be inferred evenbefore OPC is carried out. Simple line-space patterns at a most criticalpitch are often chosen. The anchor feature is used to experimentally setthe exposure dose when a mask is first printed, and the impact of thiseventual dose-centering operation is preferably taken into account whenthe compensated mask features are designed by OPC.

The process/tool then next executes, in accordance with aspects of thisinvention, a procedure that determines the coherent system set andloxicoherent system set which together produce intensity patterns thatapproximately match the images produced by the partially coherentlithographic system. This procedure is represented by a group of stepsthat are designated in FIG. 30A as Step Group 1200; these steps areexplained in more detail below in relation to FIG. 30B. The coherent andloxicoherent system sets provided by Step Group 1200 allow the imagesproduced by trial mask patterns to be determined more rapidly and/oraccurately than is possible in the prior art.

At Block 1102 the tool, in accordance with exemplary embodiments of thisinvention, begins processing the mask regions that are queued in aninput stream. This can be accomplished by applying the steps in Blocks1102A-1102E to each input mask region in sequence, though moresophisticated embodiments can process multiple regions in parallel. Mostoften a plurality of processors will participate in the execution ofthese steps even where the processing of a single mask region isconcerned, with each processor providing the dimensionally compensatedoutput patterns for a single frame of the region, but with theprocessors sharing data in order to “stitch” the output patterns acrossoverlapped guard bands, i.e. to reconcile any dimensional divergencesthat arise from the different proximity cutoffs that are entailed by thedivision of the region into different frames whose overlap is onlyfinite. As discussed, the span of any single calculated image isgenerally limited to a single frame, and because the calculations withinthe frame do not quite achieve linear scaling, the frame size istypically held below 5 microns or 10 microns in order to avoidexcessively long runtimes, i.e. the frame size is usually kept a fewtimes smaller than a typical mask region.

At Block 1102A a queued mask region is received from the input stream,and at Block 1102B the region is split into frames. This is discussed inmore detail below. In a basic implementation, each frame contains aninner core, whose typical size might be a few microns, with these frameinner cores being laid out in a grid that evenly divides the maskregion, and with each point in the mask region falling within a singlegrid box, and thus being located within a single frame inner core.However, each frame inner core is surrounded by a guard band, of widthe.g. 1 or 2 microns, that overlaps the inner core of the adjacent frame,so that the full frames overlap, with some points in the inner core ofone frame also falling within the guard band of the adjacent frame,namely those points which are separated from the inner frame boundary byno more than the so-called guard band distance. Here the term guard banddistance refers to the width of the guard band, which may be chosenequal to the optical ambit.

At Block 1102C the frames are sent to separate (but communicating)processors to generate the output shapes for each frame. The number ofprocessors handling the frames for each region is denoted F; typically Fmight be in the range of 4 to 16. The processors operate in parallel,but some frame processing can proceed sequentially if F is smaller thanthe frame count. Each processor creates output mask shapes by executingthe procedure designated as Step Group 1300, to be discussed in relationto FIG. 30C. This procedure includes a repeated step 1304E in whichguard band data is communicated between processors, as indicated in FIG.30A using dotted line arrows. Further details are provided below.

Though not shown in FIG. 30A, the number of processors employed by thetool in accordance with this invention will typically be much largerthan F, e.g. the tool might use 1024 processors, with only e.g. F=8being used in the single flow shown in FIG. 30A. The invention is ableto achieve this increase in employed computational resource byprocessing multiple mask regions in parallel, with each region beingprocessed, e.g., according to the FIG. 30A flow.

At Block 1102D the inner cores of the frames which are output from eachof the F processors are collected, and then re-tiled per the regiongridding to form an output mask region, and in Block 1102E this outputregion is transferred to the output database or stream.

It will be clear to those skilled in the art that other standard processsteps can be applied to the dimensionally compensated mask regions inthe output database in order to fabricate a finished lithographic mask.Since the regions are large compared to the optical ambit it can beuseful to exploit redundancy in the mask layout, taking advantage of thefact that integrated circuit designs often contain many repeatedregions. This is sometimes referred to in the art as exploiting layouthierarchy. As a given mask region is repeatedly inserted into differentparts of the overall layout, the content of the neighboring regions thatsurround each repeat will generally be different in each insertion, andthere are known methods for adjusting the dimensional compensation ofshapes near the border of each repeat to accommodate the varyingproximity impacts.

Since the mask regions in the output database are large compared to theoptical ambit, efficiency is not at a premium when carrying out shapereconciliation along the borders of deployed regions. However, it willbe clear to those skilled in the art that the invention can be adaptedto perform this reconciliation with greater efficiency or accuracythrough its use of loxicoherent systems.

FIG. 30B conveys in further detail the actions carried out by the toolin accordance with embodiments of this invention to obtain the coherentsystem set and loxicoherent system set that are used to match the TCC ofthe lithographic imaging system, providing in particular a flow diagramof the procedure referred to in FIG. 30A as Step Group 1200.

At Block 1204 of Step Group 1200, the invention determines a fullbilinear TCC for the specified imaging system over a doubled maskdomain.

The concept of the “doubled mask domain” has been explained above.Equation 1A, which expresses the basic behavior of partially coherentimaging, shows that points in the mask pattern m(x) contribute pairwiseto the image; in other words, the image intensity at a given point isnot made up of a sum of contributions from all mask locations that arewithin resolution range of the point in question, but rather theintensity is given by a sum of contributions from all pairs of pointsthat are (both) within resolution range of the image point. Thisphysical behavior is reflected in the double integration over the maskdomain, which essentially sums over all pairs of points (x₁,x₂) on themask (or, in practice, all pairs of points within the simulation field).Such an interaction can be regarded as an augmentation or expansion ofthe mask into what has been referred to as a doubled domain or bilineardomain.

At Block 1206 the tool in accordance with embodiments of this inventiondecomposes the TCC as a sum of coherent systems that are separated alongeach mask axis of the bilinear domain, and identifies what will bereferred to as the preferred coherent kernels (e.g., eigenfunctions orother chosen lens aperture functions to carry-out coherentdecomposition).

Summarized in computational terms, the spatial-domain TCC (referred toherein as the tcc) represents the weights with which the contributionfrom the doubled mask content at all pairs of neighboring points aresummed in order to generate the intensity at a given point. Thediscretized tcc can be written as a matrix, where different rows andcolumns represent mask points at different relative distances from thegiven point, with the rows and columns both being involved because ofthe above pairwise weighting of the contributions. The eigenfunctions ofthis tcc matrix are ordinarily chosen as the normalized OCS kernels, andmust be properly scaled through multiplication by the square root of theassociated eigenvalue. The most dominant kernels are those with thelargest eigenvalues.

It is known in the art that, even though choosing the dominanteigenfunctions of the TCC operator as coherent Mercer kernels willproduce the most rapid OCS-based extraction of the TCC possible, thusyielding the most broadly accurate OCS-based imaging in general terms,one can sometimes obtain more accurate images of the narrow set ofcritical patterns in a particular IC level by making a more specializedor tailored choice of coherent kernels, i.e. tailoring the lensapertures of the employed coherent systems to better match particularmask content of special importance. For example, Li et al., in U.S. Pat.No. 7,933,471, “Method and system for correlating physical modelrepresentation to pattern layout,” show how to form coherent kernelsthat are specialized to particular pattern content by linearly combiningwith optimal coefficients the eigenfunctions of the TCC. In a relatedreference, “Kernel Count Reduction in Model Based Optical ProximityCorrection Process Models,” Jpn. J. Appl. Phys. 48,6S (2009), Li et al.show how to choose coherent kernels that map particularly well tosliver-like pattern-changes that are made when mask fragments are finelyadjusted during OPC. Since these specialized systems continue to beMercer terms based on (now specialized) kernels that are coherent, theyare very different from the loxicoherent or rotated systems of thepresent invention; e.g. in this invention the image is not approximatedas a pure sum of coherent system contributions, and thus the image isnot calculated as a simple sum of squared convolutions of kernels withthe mask, as it is with the specialized coherent kernels of the priorart. In most embodiments the loxicoherent systems do contain constituentcoherent systems, but the loxicoherent system output is obtained byusing the output of the constituent coherent system as an input to aconstituent incoherent system. A related point that bears mentioning isthat Li et al. use the term “rotation matrix” to refer to the matrix ofcoefficients for the linear combinations of OCS kernels that they use astargeted coherent kernels; they choose this term because theircoefficient matrix must be an orthonormal matrix, and in someconventions “rotation matrix” is a synonym for an orthonormal matrix.However, this use of the term “rotation” has no connection with therotated or slanted axes on which the novel kernels of the presentinvention's rotated and loxicoherent systems are separated along. Theselatter axes are rotated in a direction within the doubled Hopkins domainthat is not orthogonal to the main f₁ and f₂ (frequency domain) axes ofthe mask. In contrast, the specialized coherent kernels of Li et al.remain functions of the main mask coordinates (i.e., the axes of theircoherent kernels are not rotated), and in their case the term “rotation”simply indicates that the squares of the coefficients that combine theTCC eigenfunctions to form each specialized coherent kernel must sum to1, and that the sets of coefficients for different specialized coherentkernels must be orthogonal to one another, thereby making the conversionmatrix orthonormal. This in turn does mean that the Li et al. conversionmatrix can be described as a rotation matrix, but only under a meaningof the term “rotation” which is quite distinct from that which describesthe axes of certain kernels of this invention.

However, even though the loxicoherent kernels of the present inventionhave no similarity to the prior art specialized or targeted coherentkernels, joint employment of loxicoherent kernels and targeted coherentkernels is nonetheless possible, i.e., the two are quite compatible withone another. In other words, since specialized coherent kernels continueto be subject to the limitations described in FIGS. 3, 4, and 7, thepresent invention and its advantages are fully consistent with use ofthese specialized coherent kernels amongst the coherent systems (N innumber) which form the TCC^((r)) from which loxicoherent systems areextracted in accordance with this invention. To allow for thispossibility we will refer to the N coherent systems in the coherentsystem set as “the preferred coherent systems”, whose kernels maycomprise, for example, the dominant eigenfunctions of the TCC operator(meaning per customary parlance the eigenvectors associated with thelargest eigenvalues, as in standard OCS); however, the preferredcoherent kernels can also include customized coherent systems that maybe designed by known methods to correlate strongly with critical maskcontent.

At Block 1208, and in accordance with an aspect of this invention, theresidual TCC is formed by removing the preferred coherent systems fromthe full TCC. This may be done by using a truncated Mercer series toform an approximate TCC, and then subtracting this approximate TCC fromthe full TCC.

At Block 1210 the tool in accordance with embodiments of this inventiondecomposes the residual TCC as a sum of multiplied lower-dimensionedkernels that are separated along axes which are rotated between the maskcontent axes in the doubled domain.

At Block 1212 the tool in accordance with embodiments of this inventiondecomposes at least one low-dimensioned kernel lying within thedoubled-domain in the mean-frequency direction into a product of maskfilters, thus determining a constituent coherent system.

At Block 1214 the tool in accordance with embodiments of this inventionselects as an intensity kernel at least one low-dimensioned kernel lyingalong the doubled-domain axis in the difference-frequency direction,thereby determining a constituent incoherent system.

Referring next to FIG. 30C, a flow diagram is provided for the procedurereferred to in FIG. 30A as Step Group 1300, this being in particular theprocedure with which the tool in accordance with embodiments of thisinvention determines a set of dimensionally compensated shapes within aparticular mask frame. In most embodiments Step Group 1300 is separatelyexecuted for each frame, typically on multiple separate processors thatprocess multiple frames simultaneously.

At Block 1302 each processor handling the patterns within a particularframe generates a starting mask, meaning a starting set of shapes withinthe frame, including adjustable edge fragments for the mask features tobe projected and printed, as well as assisting features, and in additiondetermines the target positions where the edges of the projectedintegrated circuit features should be printed.

With specific regard to the adjustable “edge fragments” and “assistingfeatures”, it should be noted that the “edge fragments” and the “maskfragments” (referred to below with respect to Blocks 1304C and 1304D)both refer here to mask edge fragments. Block 1302 may be viewedbasically as a conventional OPC step. Fragmentation of mask edgesessentially creates the “levers” that OPC uses to control printedshapes, based on the following considerations: IC design patterns areusually Manhattan polygons whose edge lengths are comparable in size tothe lens resolution. Because the lens resolution response function haslong tails, the position of the contour that is printed from an imagedmask polygon will be distorted in a complicated way by optical proximityeffects, with the distortion along a printed edge roughly taking theform of e.g. a retraction, protrusion, or wavering of the print contourin and out from the design edge (target edge), due to the varyingproximity contributions from the features that neighbor different partsof the edge with differing proximity dispositions. OPC uses maskpolygons that generally resemble the target polygons, but with edgesthat are broken up into finer “fragments” or segmented sections. OPCthen pulls each adjustable fragment in or out in such a way as to pullthe adjacent wavering print contour back into alignment with the targetedge. In basic OPC implementations the fragments are “Manhattan”, i.e.oriented along the x or y axes of the integrated circuit design, and theposition of each adjusted fragment is treated algorithmically as a“lever” that controls the position of the (locally quasi-parallel) printcontour along the normal to the fragment midpoint, with the fragmentposition only being shifted along this normal (i.e. not in theperpendicular direction), and with the length of the fragment being keptfixed. With the exception of corners, such an implementation may onlyallow adjustment of (roughly speaking) every other segment within eachfragmented edge of a Manhattan mask polygon, namely those edge segmentswhich are roughly parallel to the adjacent print contour, with thelength and position of the edges that connect these adjustable fragmentsthen being fully determined once the adjustment values are chosen.

The position adjustments made during OPC thus take the form ofintroduced retractions or protrusions of fragments that serve tocompensate the tendency of the adjacent print contour to protrude orretract, which if left uncompensated would leave the printed featurewith an improper dimension along the cross-section. The print contourwill successfully trace its desired target positions and thus provideproper dimensions once a suitable set of position adjustments have beenapplied to the mask fragments. Such a properly adjusted mask is referredto as dimensionally compensated, and in 2D the dimensionally compensatedpatterns may have distinctly different shapes from the target patterns.

In a basic OPC implementation, the proper adjustments are arrived at byan iterated feedback methodology (e.g. per Block 1304, to be discussed),meaning that the trial adjustment which is made (during one iteration)to a given fragment is chosen purely on the basis of the shortfall orexcess in the local position or intensity of the contour that isimmediately adjacent to that fragment. For example, an adjustment may bemade if there is a non-zero separation between two particularintersection points along the normal to the midpoint of the givenfragment, namely the nearby print contour location where the fragmentnormal intersects the print contour, and the location where the fragmentnormal intersects the target contour. Alternatively, an adjustment maybe made if the intensity at the intersection of the fragment normal withthe target contour exceeds or falls short of the intensity at the printcontour of the anchoring feature. (The print contour location may beadjusted to take into account the offset predicted by anintensity-driven resist model, as is well-known in the art. Theeffective intensity at the target edge may be similarly adjusted to takeresist effects into account, as is also well-known in the art.) The(generally weaker) impact of the trial adjustment on other neighboringportions of the print contour is not considered in a direct way by sucha feedback scheme, and as a result the printed dimensions will not befully compensated by the adjustments made during a single iteration.However, the controlling fragment adjacent to a particular print edgewill generally exert the dominant influence on the position of thatprinted edge, allowing the feedback methodology to properly converge allprint positions within tolerance after several iterations, e.g. withinabout 10 iterations. Typical tolerances might be of order 0.1nanometers. (This description has been highly abbreviated for brevity;it will be clear to those skilled in the art that many other prior artOPC techniques and features can usefully be incorporated into theinvention's procedure. These well-known techniques allow OPC toconsider, e.g., slope information, non-Manhattan geometries, a pluralityof process conditions, resist effects, and mask-making constraints;however these techniques and features are largely independent of whetherstandard OCS or the novel loxicoherent systems of the invention are usedto calculate intensities. Resist effects in particular can benumerically significant, but standard pre-calibrated models of resistbehavior have been developed to take resist effects into account duringOPC; this is done by essentially determining a local shift in the doselevel at which the resist develops out [or determining an equivalenteffective intensity shift], and the input to these standard models isthe exposing intensity pattern itself, which the invention determinesmore efficiently than prior art tools.)

In a preferred embodiment, convergence is made more monotonic by dampingthe adjustment, e.g., in a single iteration each adjustable edgefragment might be shifted by a lesser amount than that which isestimated to fully correct the adjacent print contour. In general, theratio of the induced shift in the position of a printed contour and thedriving shift in the adjacent mask edge which causes the contour shift,is known as the Mask Error Enhancement Factor, or MEEF, and MEEF may beestimated for particular edge fragments by sampling, or by tracking theshifts observed in previous iterations. Stability of the convergencebehavior through the course of iterations can be improved by limitingthe adjustments that are made in any single iteration to, e.g., ½ thevalues which would fully correct the remaining errors in print contourposition according to the best available estimates of the local MEEF,i.e. convergence behavior is improved by damping the adjustments.

It is impossible for uncompensated wavering of the (unconverged) printcontour to oscillate with great rapidity within the image plane, sincethe exposing image is bandlimited. (This band limit applies to opticalproximity effects for which compensation is possible; it does not applyto stochastic effects in the resist itself.) For this reason the densityof control points will usually be adequate as long as the edge fragmentsare moderately shorter than the lens resolution, e.g. a few timesshorter. On the other hand, the dominance of the controlling fragmentscan be unduly reduced if the fragment lengths are made overly fine, andthis will slow convergence. Rules are known in the art for appropriatelychoosing the fragment lengths and positions, and these so-calledfragmentation rules are specified during step 1100. The rules may beapplied to the patterns within each frame inner core during step 1302.Alternatively, the patterns in the input queue may be pre-fragmentedusing the same standard methods.

Most commonly the mask patterns in the input queue are initialized tothe target patterns, thus providing block 1302 with target printpositions. In other cases, so-called retargeting rules are used to makechanges in the specified target print positions, for example requiringthat isolated lines or equi-spaced lines of particular pitch be printedwith specified biases. Retargeting rules may be input in step 1100, andmay be applied within the individual guard-banded frames during step1302.

The “assist features” deployed during block 1302, also referred to asassisting features or SRAFs (Sub-Resolution Assist Features), arewell-known in the art. Broadly speaking, assist features are used tomitigate an aspect of printing features near the limit of resolutionwherein the pattern-dependence of the not-fully-avoidable deteriorationin image quality is increased, due to the fact that only the coarseimage harmonics fit within the lens bandwidth. This pattern-dependentvariability also increases when customized illumination directionaldistributions (referred to as “sources” for short) are used to increasethe upper limit on the density of patterns that can be resolved, andwith customized sources the variation in print quality becomesparticularly pronounced between semi-isolated features and features thatare laid out in a high density. Typically the source design is chosen insuch a way that the high density features have acceptable depth offocus, with the depth of focus for semi-isolated features remainingquite small. The assist features that mitigate this remaining focussensitivity are dummy mask features that are laid out adjacent tosemi-isolated features with a density and periodicity that approximatesthe density/periodicity of e.g. the most critical dense features beingprinted, with each assisting feature usually being kept too small tofully expose the resist at its location, i.e. assists are generally keptsub-resolution in size, so that they don't print as artifacts. It shouldbe noted that there are variant procedures in which the assist featuresare allowed to print because other masking steps will be employed infabricating the IC level that can be used to remove the printedartifacts. Even non-printing assists will, however, form a sub-thresholdspot or line-segment of light in the image, and these adjacent spacedout pulses of light, though sub-threshold, will interfere in such a wayas to cause the semi-isolated feature to print in a manner more closelyresembling the printing of dense features, thus reducing the variabilitybetween features, and allowing other RETS to be more narrowly targeted.Assist features are generally not adjusted during OPC, but are laid outin fixed positions within the starting mask. The assists may be providedin the input, or they may be deployed during Block 1302 using so-calledassist rules that may be supplied during step 1100.

At Block 1304 the processor handling the patterns within a frameassigned to it adjusts the mask fragments (sometimes simply referred toas edges for brevity) within the frame by repeatedly cycling (iterating)the steps of:

Block 1304A: determining loxicoherent system contributions to the imageintensity at target edge positions by applying the intensity kernels tosquared mask transmissions that have been filtered by the mask filters;

Block 1304B: determining the image intensity at target edge positions byadding the loxicoherent contributions to the sum of intensities from thepreferred coherent systems;

Block 1304C: moving mask fragments adjacent to target edge positionswhose intensity is lower than the intensity at the edge of the anchoringfeature in a direction towards the ‘darker’ side of the adjacent targetedge;

Block 1304D: moving mask fragments adjacent to target edge positionswhose intensity is higher than the intensity at the edge of theanchoring feature in a direction towards the ‘brighter’ side of theadjacent target edge;

Block 1304E: transferring to other processors the iterated positions offragments within the guard band of the frame being processed, inparticular transferring this data to other processors which are handlingadjacent frames that are overlapped by this guard band; then usingposition data from the guard bands of other frames that have similarlybeen transferred from the adjacent-frame processors to unify andharmonize the positions of fragments in the exterior guard band of theframe being processed before commencing the next iteration cycle; and

Block 1304F: terminating the adjustment cycles when the intensities atall target edge positions match that of the anchoring feature to withina tolerance.

Block 1304 is repeated several times across multiple iterations (e.g.10), and across multiple processors handling different frames (e.g. 16).Each iteration refines the results of the previous iteration, includingprevious iteration results from adjacent overlapped frames via Bock1304E. Further details on the handling of data from adjacent overlappedframes may be understood with reference to FIG. 31. In particular, FIG.31A shows, in schematic form, a mask area where four adjacent framesintersect in a corner. Only the corner regions of the frames are shown,with the inner core portions of the frames being indicated in highlyschematic fashion by the four letters A, B, C, and D. In addition, forone of the frames (frame A), the corner portion of the boundary for thefull frame, including the guard band, is shown as a thick line. Thisboundary extends into the neighboring frames, since the guard band ofeach frame overlaps the inner cores of adjacent frames by a uniformdistance, which might in practice be chosen as 1 micron or 2 microns,e.g. the overlap might be set to the OD.

When the processor handling frame A makes an iterative adjustment in thepositions of the mask shape fragments contained within its frame (duringBlocks 1304C and 1304D), the shapes from adjacent frames which areoverlapped by the guard band of frame A could, in a basic embodiment, beamong those receiving an adjustment, since the guard band is part of theframe. As a specific example, those shapes labeled schematically as “B”which lie inside the thick-line-delineated boundary of frame A (thisboundary being shown along the lower right corner of frame A) could beadjusted by the frame A processor along with the other shapes withinframe A. Since these B shapes fall within the inner core of frame B,they will also receive an independent adjustment by the processorhandling frame B. More precisely, when the frames are distributed toseparate processors in Block 1102C, the shapes within the guard bandregions where frames A and B overlap will be sent (essentially ascopies) to two separate processors for adjustment (as well as toadditional processors in the extreme corner regions where four framesoverlap). During the adjustment process the optical impact ofneighboring patterns is accounted for in Block 1304B, but thedescription of these neighboring patterns that is available to eachprocessor is incomplete for patterns in the exterior guard bands; inother words, the guard band fulfils its purpose of providing opticalcontext for patterns in the frame inner core, but the guard band doesnot itself have a guard band. Periodic boundary conditions are typicallyemployed during OPC (as a consequence of using FFT-based convolutions),meaning that the neighboring environment for e.g. the “B” patterns inthe exterior guard band of frame A includes a “fictional” repeat offrame A to the right of the thick-line frame boundary shown in FIG. 31A.Alternatively, the image may be obtained using only the patterns presentwithin the frame, meaning that the exterior region is effectivelytreated as empty. In general, the neighborhood environment for, e.g.,the B patterns lying within the guard band of frame A will not becorrectly represented in the region to the right of the frame boundary(as far as the processor assigned to frame A is concerned). Thus, aniterated adjustment made by the frame A processor to these B patternswould be less accurate. However, as discussed above, the adjustmentsmade during each iteration are preferably damped, so the inaccuracyintroduced into these B patterns relative to their previous position isfairly modest. This means that these adjusted B patterns are able toserve as a reasonably faithful representation of the neighborhoodenvironment when the A patterns on their interior side are beingadjusted by the frame A processor, both in terms of their opticalimpact, and in accounting for any mask manufacturability limits that mayarise at the inner core boundary. In other words, the inclusion of aguard band containing B patterns allows the frame A processor to make areasonably accurate adjustment to the inner core patterns in frame Athat are adjacent inwardly to the frame B patterns, even though theframe A processor is not able to adjust the overlapped B patternsthemselves very accurately.

In alternative embodiments the frame A processor only makes adjustmentsto the frame A inner core patterns during Blocks 1304C and 1304D, i.e.the exterior guard band patterns are left unadjusted during these steps,but in such embodiments a similar conclusion can still be drawn; theomitted adjustments in the exterior portion of the guard band will notbe of large magnitude in any single iteration, meaning that theunadjusted guard band patterns in the B overlap region can still providereasonable context for the adjacent frame A inner core patterns, atleast for the first iteration in which guard band adjustment is omitted(and, as will be seen, proper guard band adjustment can still beprovided before the next iteration commences).

The above situation is of course reversed for the frame B processor;that processor cannot generally provide an accurate adjustment for theframe A inner core patterns that are within its guard band, but the Binner core patterns that are within the guard band of frame A (and thusnot accurately handled by the frame A processor) can be handledreasonably well by the frame B processor. Given this complementaryaccuracy in coverage, a reasonable strategy for the Block 1304E overlapreconciliation is to simply swap the iteration results for exteriorguard band locations in each frame, replacing them with thecorresponding inner core results from adjacent frames. In other words,since the processor for frame B can obtain more accurate adjustmentsthan the frame A processor for those B patterns that are also used as aguard band by the frame A processor, it is reasonable for the Aprocessor to replace the edge positions of these B pattern edgefragments, when commencing the next iteration, by the edge positionsthat were calculated by the B processor during the previous iteration,these B results being received by the A processor during the Block 1304Ereconciliation step. In general, each processor replaces the fragmentpositions that it obtains for patterns within the exterior guard band ofits frame with the (generally more suitable) positions given to thesepatterns as part of the inner core of an adjacent frame. Similarly, theprocessor also transfers to other processors the edge positions offragments within the inner core of its frame that also fall within theguard bands of other frames.

With this simple swap scheme an abrupt change is made to patternsoutside the frame inner core, while patterns inside the inner core areleft entirely unchanged. More sophisticated schemes can be used in whichthe boundary reconciliation changes are spatially smoothed. One suchscheme may be understood with reference to FIG. 31B, which shows thesame corner region between four frames as FIG. 31A. In the FIG. 31Bembodiment the frame boundaries have been extended to increase theoverlap with the inner cores of adjacent frames, as may be seen from theframe A outer boundary, which is again indicated in its corner portionwith a thick solid line. As with the simple swap scheme of FIG. 31A, theFIG. 31B smoothing scheme for boundary reconciliation can leaveunchanged (during Block 1304E) the edge positions calculated by theprocessors for shapes in their frame inner cores. For example, theboundary of the frame A inner core is shown in FIG. 31B as a thickdashed line, and patterns inside this boundary can be left unchanged bythe frame A processor during reconciliation step 1304E. However,reconciliation changes will in general be made by the frame A processorto guard band patterns that lie outside the frame A inner core, using alinear interpolation that blends two categories of data; first, theprevious-iteration edge positions that the frame A processor itselfobtains during application of steps 1304A through 1304D to the fullframe A (including exterior guard band regions), and second, theprevious-iteration edge positions that the frame A processor receivesfrom the processors of adjacent frames during step 1304E. The specificmode of interpolation can be chosen based on the overlapping framesinvolved. For example, the exterior guard band for frame A may bedivided into 8 regions involving different overlap combinations, ofwhich 3 are shown (or partially shown) in FIG. 31B. In the strip-likeregion (partially shown) where only frames A and B overlap (outside theframe A inner core), the interpolation weight given to the processor Aresults may be ramped linearly from 100% on the left side of the stripto 0 on the right, with the weight given to the B results being rampedin complementary fashion, i.e. from 0 to 100% left-to-right. Similarly,the fragment positions used in the (partially shown) strip-like guardband portion where only frames A and C overlap may be smoothly mergedbetween the frame A and frame C results by using complementaryinterpolation weights that ramp linearly from top to bottom of thestrip.

In corner portions of the guard band where four frames overlap (e.g.,the square-shaped corner portion of the frame A guard band whose edgesare delineated in FIG. 31B with thick solid lines and thin dashedlines), the fragment positions can be set using a bilinear interpolationof the results from the four overlapping frames, with each interpolationweight being the product of a linear ramp for the x coordinate and alinear ramp for the y coordinate. For example, the result from frame Amay be given a weight that is the product of a first factor that rampsfrom 100% to 0 as the x coordinate of the fragment midpoint in questionvaries from the left edge of the corner region to the right edge, and asecond factor that ramps from 100% to 0 as the y coordinate varies fromtop to bottom of the corner region. Similarly, the weight given to theframe B result can be a product of linear terms that ramp from 0 to 100%left-to-right, and 100% to 0 top-to-bottom. The weighting system extendsin the obvious way to the other frames that overlap in this(lower-right) guard band corner of frame A, and likewise to the othercorners of the frame A guard band, and further to the guard bands ofother frames, as does the weighting scheme for handling the strip-likeportions of exterior guard bands where two frames overlap. This smoothedreconciliation of guard band data during Block 1304E provides continuitywith adjacent frames while allowing the guard bands to continueproviding suitable context for the inner cores, even as the frames arebeing processed (largely) in parallel.

Resist effects can be taken into account with an additional step betweenBlocks 1304B and 1304C in which one of the standard resist models usedin OPC is applied, as will be clear to those skilled in the art. Theseresist models use the exposing intensity pattern (e.g. as obtainedduring Block 1304B) as input, and provide as output e.g. an effective(phenomenological) change in the local intensity at each feature edge,such that when the revised intensity pattern is thresholded at theexposure dose required by the resist, the resulting revised dose contourwill accurately reflect the deviations of the developed resist edgecontour from the physical dose tracking response of an ideal resist. Theedge adjustments in Blocks 1304C and 1304D can make use of the effectiveintensity as modified by the resist model, rather than referencing thetrue optical intensity. Some resist models ostensibly determine aneffective change in the local development threshold instead of the localintensity level, but for the execution of Blocks 1304C and 1304D thisonly amounts to a sign change in the revision. As is well-known, thestandard resist models determine the dose or intensity adjustment ateach feature edge by applying a model function whose terms can beinterpreted as “traits” extracted from the local image; for example,these traits may include the local slope of the intensity along acutline that crosses the feature edge, or the maximum or minimum valuesof the intensity along such a cutline, or the values taken on at thefeature edge by convolutions of the intensity with regression kernels.The standard functional forms for the resist model are generallyregression polynomials or modified polynomials whose coefficients andparameters are determined in a data fitting step undertaken before OPCcommences, i.e. a regression step in which the resist model parametersare fit to e.g. a few thousand measured dimensions of a diverse set ofcalibration patterns exposed in resist under different dose and focusconditions.

The adjustments in Blocks 1304C and 1304D should also respect standardprecautions to ensure that the dimensionally compensated aperture shapescan actually be manufactured on the mask, which may mean accepting animperfect dimensional compensation. A common approach is to check eachadjustment in fragment position against so-called mask manufacturabilityrules, which in the simplest case specify minimum separations that mustbe maintained between any two facing edges of an opaque mask feature, orbetween the feature edges at opposite sides of an aperture shape formedon the mask. The adjustments in Blocks 1304C and 1304D should thus beclipped to maintain the minimum spacings imposed by thesemanufacturability rules, and the termination criterion expressed inBlock 1304F should be understood to include an exception for edges wherecomplete equalization of the target edge intensity to the common dose ofthe anchor contour is blocked by mask manufacturability constraints.

Block 1304A allows for the use of a plurality of loxicoherent systems,and the set of loxicoherent systems employed by the invention (or thesingle loxicoherent system, if only one is used) will be referred to asthe loxicoherent system set. In accordance with an aspect of thisinvention the presence of even a single loxicoherent kernel (requiring 2FFTs to apply) has been found to remove most of the 1D error; this isachieved by the removal of the fin (along the diagonal), with resultingbenefit as has been illustrated in e.g. FIG. 28.

The use of the first loxicoherent kernel provides a strong incrementalbenefit. As was discussed above, the prior art OCS method is known toprovide the best possible technique to approximate the imaging operator(i.e. the TCC) as a sum of coherent systems, although with specializedmask content it can be useful to apply specialized OCS kernels, as isnow well-known. This optimality property might seem to contradict thestrong advantage seen from applying loxicoherent systems. However, whilethe optimality results known in the art establish that OCS providesoptimum kernels for matching the TCC using Mercer terms having the formΨ(f₁)Ψ*(f₂), these results do not prove the superiority of OCS over,e.g., expansions with terms according to the invention having the form

(f){tilde over (T)}(Δf) or T′(f₁) T′*(f₂) T″(Δf). Explained in morephysical terms, the optimality proofs in the literature show that theeigenfunctions of the TCC operator provide the best possible lensapertures for a set of coherent systems whose images are superposed(summed) to match a partially coherent lens system, but they do notaddress the performance of the novel loxicoherent systems employed bythe invention (or even indicate awareness of such novel compound systemsin any way), and these loxicoherent systems have been shown herein to bestrongly useful in image calculations. Even though OCS/Mercer kernelsmay be preferable for the first few terms of the decomposition, itquickly becomes preferable to add at least one loxicoherent system inaccordance with the embodiments of this invention.

The OCS terms have difficulties at the Δf=0 crease because at theselocations ∂²TCC/∂Δf² must become essentially infinite, even while∂²TCC/∂f ² remains of moderate magnitude. Strictly speaking, theassumption that imaging is isoplanatic over the OD is not perfectlyaccurate, and the notion of an infinitely sharp TCC crease breaks downat some point. However, as a practical matter, field-dependentlithographic aberrations are quite small to begin with, and the OD isvery small compared to the full lens field (e.g. a few microns versustens of millimeters), so TCC creases may be considered arbitrarily sharpat any scale that is relevant to OPC.

As has been discussed in connection with FIG. 11, one can readilyprovide the approximated TCC with very different curvatures along the Δfand f meridians by using terms of the form T_(Rotated)≡

(f){tilde over (T)}(Δf), with each of the distinct constituent kernelsbeing given very different curvatures, whereas, per FIG. 7, conventionalMercer terms have curvatures in these two meridians that tend to bebroadly similar to order of magnitude. Use of rotated axes thus allowsthe approximated TCC to better match the sharp crease in the true TCCthat conventional OCS kernels can only slowly extract. The sameconclusion of superior compatibility applies to the T′(f₁)T′*(f₂)T″(Δf)loxicoherent system decomposition. The FIG. 30B procedure in effectassigns {tilde over (T)}(Δf) an effectively infinite sharpnesscorresponding to ˜1 pixel of the simulation grid.

Reference in this regard can be made to the plot shown in FIG. 32 thatillustrates a T″ filter kernel, along with a t′ spatial domain kernel.FIG. 32 is complementary to FIG. 23, with the right side of each plotshowing frequency domain kernels. The sharp peak at the origin in the T″frequency domain kernel in FIG. 32 is apparent (C-quad test caseintroduced in FIG. 9). In contrast, the T′ filter kernel shown in FIG.23 (right-side plot) has only a moderate curvature.

Although the first loxicoherent system can extract large portions of theresidual TCC error, the ideal asymptotic limit in which TCC^((r)) isentirely extracted by this one system cannot be fully realized inpractice, and the first system will still leave some residual TCC error,as has been illustrated in FIG. 24. This remaining TCC error can beregarded as a new TCC^((r)) from which additional loxicoherent systemscan be extracted. However, additional mathematical considerations areinvolved in these extractions, since it is no longer appropriate tocarry out step 1212 (where the constituent coherent system aperture T′is determined) of the FIG. 30B procedure using FIG. 22G in unmodifiedform. As with the first loxicoherent system, one may once again extractnew rotated systems (i.e. of the FIG. 11A form) from the new TCC^((r))by using the eigenfunctions of operators Q and Z in FIGS. 12F and 12G,but FIG. 22G no longer provides an optimal route for extracting newloxicoherent kernels (in particular, the constituent coherent systemaperture, also referred to as a mask filter) from these rotated systems.The difficulty is that FIG. 22F (and thus FIG. 22G) is contingent on thepresence in TCC^((r)) of a dominating fin along Δf=0, and this fin isremoved once the first loxicoherent kernel is extracted, as may be seenby comparing FIG. 24 to FIG. 14.

It is, of course, straightforward to extract valid constituent coherentkernels from the T_(Rotated) kernels by diagonalizing the

kernel into a Mercer series in its eigenfunctions, which would achievethe form specified in the last step of FIG. 20A, and then setting T″equal to {tilde over (T)}. However, each such loxicoherent kernel wouldtend to be highly sub-optimal. This can be understood by noting thatwhen

is treated as an operator in the doubly-dimensioned Hopkins space, itwill depend on its two arguments (the f₁ and f₂ coordinates) only in thecombination f=(f₁+f₂)/2. It follows that when

is inverse Fourier transformed under complex operator conventions(meaning that the transform in the f₂ argument uses a conjugatedexponent), the resulting spatial domain function

will be Toeplitz, in the sense that it will depend only on x₁−x₂, sothat its gridded form will be a Toeplitz matrix. In general the Mercerexpansion (diagonalization) of such a matrix will only converge slowly,and this slow convergence will be replicated back in the frequencydomain, i.e. in the diagonalization of

$\overset{︵}{T}\;{\left( \frac{f_{1} + f_{2}}{2} \right).}$This in turn implies that a very large number of terms K would need tobe used in the right-hand side of FIG. 20A in order to get a closeapproximation of the first new rotated system, where the first newrotated system referred to here is the first rotated system to beextracted from the new TCC^((r)) (i.e. the newly remaining TCC error)that is formed after the first loxicoherent system has been extracted.

However, even though direct use of eigendecomposition in the last stepin FIG. 20A would typically require an impractical number of terms K, adirect Mercer expansion of this kind can prove helpful as one element ina more sophisticated decomposition method, as will be discussed.

Before describing this and other more sophisticated decompositions, itis helpful to first consider a simpler but more straightforward methodfor extracting a second loxicoherent system (and subsequent higher-orderloxicoherent systems). While the kernels provided by this simpler methodare mildly sub-optimal in their reduction of RMS TCC^((r)), they areoften a close enough match to TCC^((r)) to provide an appreciable andquite useful accuracy improvement. This simple method is based on theexpectation that after only a single loxicoherent kernel has beenextracted to remove the Δf=0 DC fin, one may still expect to seemoderately pronounced residual ridge-like content in TCC^((r)) alongadjacent diagonally displaced contours where Δf has constant (butgenerally non-zero) magnitude. Although TCC^((r)) will at this point beextremely small (and preferably zero) where Δf is exactly 0,non-negligible “diagonal” ridges or “ripples” will often be present atlow values of Δf that closely flank the former peak of the removed finat Δf=0, as may be seen in FIG. 24. The T″ kernel of the firstloxicoherent system may broadly be interpreted as a mean cross-sectionof the Δf=0 fin, and the residual ripples that remain after this fin isremoved may be thought of as deviations from the extracted mean. Whilethese residual ridges will be small in comparison with the removed fin,they are likely to nonetheless represent the largest remaining contentin TCC^((r)) after the first loxicoherent system is extracted. They alsorepresent content at low (but non-zero) spatial frequencies, and, as wasdiscussed in connection with FIG. 26, with practical masks a strong lowfrequency peak will almost always be present in the spectralautocorrelation function of the mask patterns, and this peak tends tomake the extraction of low frequency content in TCC^((r)) more importantfor image matching accuracy than reduction of the overall RMS magnitudeof TCC^((r)).

By focusing on extraction of TCC^((r)) in the largest diagonal “ridge”that remains after the Δf=0 fin has been extracted, it is possible toextract a second loxicoherent system using much the same procedure ashas been derived above for obtaining the first loxicoherent system (e.g.obtaining the first system by using FIGS. 22G and 22J), but substituting(as a replacement for the TCC^((r))(f,f) fin peak that is used toextract T′ for the first loxicoherent system) the residual content alonga difference frequency diagonal Δf=Δf₀ that is deemed to now be dominantin the remaining TCC residual (with Δf₀≠0). Since such content typicallyshows far less relative predominance than the Δf=0 fin in the TCC^((r))from which the first loxicoherent system is extracted, we will refer tothe strongest remaining ridge as “quasi-dominant”, and more generally wewill deem any diagonal ridge with strong residual TCC content (otherthan the Δf=0 fin) to be “sub-dominant”. As a further point ofnomenclature, when a subsequent loxicoherent system is designed toextract TCC^((r)) content in the vicinity of a particulardifference-frequency “ridge”, such as a quasi-dominant ridge, the peakdifference frequency of this ridge will typically be denoted Δf₀.

The method (to be described) for extracting loxicoherent systems thatcapture this quasi-dominant content will also succeed in capturingTCC^((r)) content at other difference frequencies throughout the Hopkinsdomain; analogously with the primary loxicoherent system, the pluralityof constituent systems that are present in a higher-order loxicoherentsystem allows complete or near-complete suppression of a quasi-dominantpeak in TCC^((r)) via one constituent kernel, while also providingsimultaneous minimization of RMS TCC^((r)) over the entire doubleddomain using the other kernel. Moreover, simple iteration of the methodallows multiple loxicoherent systems to be defined in a systematic way,since extraction of a quasi-dominant ridge may be followed by a newapplication of the method to another quasi-dominant diagonal that ispresent in the TCC^((r)) that still remains. (The quasi-dominantdiagonals in the still-remaining TCC^((r)) will generally all have beenchanged somewhat, since the previously extracted system will generallyhave reduced TCC^((r)) throughout the domain.) Just as the firstloxicoherent system may be referred to as a first-order loxicoherentsystem, these succeeding loxicoherent systems may be referred to ashigher-order loxicoherent systems.

It should be noted that FIG. 22E will not be applicable in general alongΔf contours other than Δf=0, since the remaining TCC^((r)) after thefirst system is extracted will usually be roughly balanced in itsprimary real part between positive and negative residuals. Appropriatealternatives to extract loxicoherent systems beyond the first will nowbe presented. In the usual case where the quasi-dominant Δf diagonalsare of low (but non-zero) frequency, it is appropriate to applyloxicoherent systems in the specialized form of FIG. 20I, rather thanthe simpler form of FIG. 20H that suffices for the first loxicoherentsystem. For example, a single T′ kernel might be extracted for a ˜50%region (or for disjoint regions collectively comprising ˜50%) along theΔf=Δf₀ diagonal within which the real part of TCC^((r)) is positive, andanother T′ kernel extracted for regions along this quasi-dominantdiagonal wherein TCC^((r)) is largely negative, so that parameters R⁽⁺⁾and R⁽⁻⁾ would both chosen as 1 in this non-limiting example.

A useful consequence of such choices is that the logarithm of the T′kernel that is associated with a particular choice of r⁽⁺⁾ or r⁽⁻⁾ inFIG. 20I will be finite within any contiguous region of the windowfunction D, and need not cross a branch cut. In principle this allowsisolation of the nonlinear character of the condition that matches thenew loxicoherent system to TCC^((r)) along the quasi-dominant Δf₀diagonal, by use of logarithms.

Referring to FIG. 33, the condition for matching a loxicoherent systemto TCC^((r)) along a quasi-dominant diagonal Δf=Δf₀ is expressed as FIG.33A, and since FIG. 33A involves products of the unknown T′ function, itis nonlinear. However, when the log of T′ is regarded as the unknown,the matching equation becomes a linear equation, as shown in the firstline of FIG. 33B. This equation holds at all f points along the Δf₀diagonal at which TCC^((r)) has the same chosen sign, so a system oflinear simultaneous equations can be obtained by gridding the equationin f. In principle T′ can then be determined by solving this system ofsimultaneous equations, with log [T″(Δf₀)] being treated as an arbitraryconstant offset, which may be set to 0. (If TCC^((r)) is negative inthis region, T″(Δf₀) can initially be set to −1, as indicated in thelast line of FIG. 33A, but with this negative sign being considered tocancel (and be canceled by) the sign of TCC^((r)), leaving log[T″(Δf₀)]=0.) However, these simultaneous equations are extremelyill-conditioned in practice. A formulation with slightly betterconditioning is obtained by recognizing that the two log [T′] termswhose sum is the log of TCC^((r)) along the Δf=Δf₀ diagonal (witharbitrary constant offset log [T″(Δf₀)], which may be set to 0) are twocopies of the same function, each being shifted from one another by Δf₀,i.e. one copy is shifted by +Δf₀/2, and the other by −Δf₀/2. Their sumcan therefore be regarded as a single copy of the log [T′] function thathas been convolved with a kernel which contains two impulses, i.e. thiskernel is the sum of a delta-function centered at +Δf₀/2 and adelta-function centered at −Δf₀/2. This conversion to a convolution isexpressed in the second line of FIG. 33B. Extraction of log [T′] fromthe known log [TCC^((r))] along a segment of the quasi-dominant diagonalcan then be treated as a deconvolution problem, to which standardregularized methods can be applied. Once T′ is determined from thequasi-dominant diagonal, T″ can be obtained using FIG. 22J.

This deconvolution method can yield reasonably well-conditionedsolutions in the case of small Δf₀, i.e. where the dominant remainingerrors in image matching involve intensity modulation at low spatialfrequencies, and the residual content that is quasi-dominant indeedusually occurs at values of Δf₀ that are small, i.e. values which lie inclose proximity to the Δf=0 fin that is removed by the firstloxicoherent system. However it is often easier to avoidill-conditioning by avoiding the logarithmic transformation, and insteadsolving the equation of FIG. 33A directly as a nonlinear optimizationproblem, e.g., minimizing the RMS error in satisfying FIG. 33A atmultiple gridded f values, potentially including weights that reflectthe generally heterogeneous frequency content of IC masks, as has beenillustrated in FIG. 27. Standard minimization algorithms can be used,and the calculation is generally quite fast because optimization takesplace over the limited dimensionality of the quasi-dominant diagonal.When Δf₀ is small, a suitable starting solution for T′ is provided bythe last line of FIG. 33C.

FIG. 33C is based on roughly the same simplification as FIG. 22F, whichwas shown to hold in the vicinity of the fin. Since the TCC^((r))content that is quasi-dominant after extraction of the firstloxicoherent system often represents content in the “foothills” of thefin that could not be exactly fit by the first T″ kernel, it is oftenthe case that Δf₀ is quite small (though not zero), which makes FIG. 33Cquite accurate. In fact, the last line of FIG. 33C may in itself providea sufficiently accurate solution for T′ along the quasi-dominantdiagonal, without any further refinement, as will be seen.

The solutions for T′ and T″ obtained using FIGS. 33A-33C may thenoptionally be further refined to maximally extract TCC^((r)) across thefull 2D or 4D domain, rather than along the quasi-dominant diagonalalone. However, it should be noted that even when T′ is only optimizedwithin the quasi-dominant diagonal, the optimization of T″ using FIG.22J considers the full 2D or 4D Hopkins domain, so that it provides adegree of reduction in TCC^((r)) throughout the full f₁,f₂ space. Inparticular, FIG. 22J provides the T″ kernel that is optimal in a leastsquares sense over the full Hopkins domain (when used in a loxicoherentsystem). Since the OCS kernels Ψ are optimal (for use in coherentsystems) under the same criterion when chosen in the standard way, itcan be seen that loxicoherent systems and OCS systems both include akernel function that provides a least squares optimal reduction inTCC^((r)) over the full Hopkins domain (for use within systems of theirrespective prescribed structures). However, unlike a coherent system,each loxicoherent system includes more than one distinct kernel, e.g. aT′ kernel in addition to a T″ kernel. In embodiments where the procedurefor determining T′ includes a step of numerical optimization, thisoptimization can optionally be carried out over the full doubled domain,or alternatively over a restricted domain, such as along aquasi-dominant diagonal. With either option, well-known optimizationmethods may be used, such as the Gauss-Newton or Levenberg-Marquardtalgorithms. These are appropriate for minimization of a least squaresobjective, such as the squared fitting error in matching the secondloxicoherent system to TCC^((r)).

Accelerated gradient methods may alternatively be used. Regularizationterms can be added to the objective to prevent imbalance in the sizes ofT′ and T″, since in the objective these two factors always occurtogether in products. If the optimization is unconstrained, animplicitly two-stage optimization can be used, with the T″ values beingobtained from FIG. 22J during optimization loops, rather than beingoptimized as variables.

When Δf₀ is large, the approximation of FIG. 33C will be inaccurate(sometimes making it unsuitable even for initializing an optimization).In addition, the most optimal form for higher order loxicoherent systemsmay differ from FIGS. 20H and 20I when Δf₀ is large. A more suitableform (which in some cases entails an additional FFT) may be understoodwith reference to FIG. 34. In particular, to obtain a suitable form fora newly added loxicoherent system in cases where the quasi-dominantfrequency Δf₀ is no longer small relative to the band limit, one canextract the system using the offset frequency coordinate system definedin FIG. 34A, also referred to as a local frequency coordinate system.The FIG. 34A local frequency variables (designated with primes) allow aloxicoherent system to be defined based on the quasi-dominant diagonalΔf₀ in much the same way that the first loxicoherent system is obtainedvia FIGS. 22A-22H from the dominant fin at Δf=0. TCC^((r)) will haveHermitian symmetry when preferred practices are followed, which meansthat it will contain symmetric quasi-dominant diagonals at both +Δf₀ and−Δf₀. However, we will first consider the case where Δf₀ represents thepositive-valued frequency difference. Toward that end, the + subscripton the primed local frequencies f′_(1,+) and f′_(2,+) indicates that thelocal coordinates in FIG. 34A refer to the positive-frequencyquasi-dominant diagonal. TCC^((r)) will have relatively large magnitudenear this diagonal, so as a working approximation we can treatf′_(1,+)−f′_(2,+) as a small quantity for frequency pairs of interest,though this assumption will be less productive than the correspondingapproximation that was made at the earlier stage of extraction where thedominant Δf=0 fin was captured by the first loxicoherent system.(However, as with the method of FIG. 22 [e.g. FIG. 22F], ourapproximation that Δf′ is small will not prevent the resultingloxicoherent system from fully extracting the quasi-dominant diagonal,while at the same time improving TCC^((r)) throughout the remainder ofthe Hopkins domain via an incoherent kernel that is least-squaresoptimal.) We further define a primed version of the TCC^((r)) functionthat expresses the residual TCC error as a function which has the primedfrequency coordinates as its arguments. In other words, we define afunction TCC₊ ^((r))′ to be a shifted version of TCC^((r)) that has beencentered at the origin of the new primed coordinate system, as expressedmathematically in FIG. 34B, where the right-hand expression then mapsthe primed arguments back to the conventionally centered frequencyarguments of the TCC, via FIG. 34A.

Since the +Δf₀ and −Δf₀ diagonals are quasi-dominant, we can efficientlyextract TCC^((r)) in each region of given sign by following much thesame procedure as was used above for the dominant Δf=0 fin, if we workin the locally rotated system. FIG. 34A shows explicitly how anefficient rotated system approximation to the local residual TCC can beexpressed in the local coordinates. For the same reasons as wereapplicable with FIG. 16B above, we can consider the local {tilde over(T)} function for the positive frequency Δf₀ diagonal to have a peakvalue of 1, with this peak occurring at Δf=Δf₀, i.e. at Δf′=0. Thismeans that the local rotated system kernel

can be determined by matching it to TCC₊ ^((r))′ along the ridge peak ofthe quasi-dominant diagonal, as shown in the first line of FIG. 34D,with this determination of

being analogous to that which was carried out in FIG. 16B for thedominant fin using unprimed coordinates. Since f′_(1,+) is numericallyclose to f′_(2,+) within this quasi-dominant diagonal region, we canapproximate TCC₊ ^((r))′ at the arithmetic mean of f′_(1,+) and f′_(2,+)along the ridge peak (i.e. with both arguments of TCC₊ ^((r))′ beinggiven this same mean value in the primed coordinates) as being veryclose in value to the geometric mean of the ridge values of TCC₊ ^((r))′at f_(1,+)′, and at f_(2,+)′. This approximation is shown as the secondline of FIG. 34D, and is analogous to that used in FIG. 22F withunprimed coordinates. Note that when TCC₊ ^((r))′ is complex-valued, thegeometric mean must be calculated in a way that ensures continuity inphases, and this may require setting

to zero near branch cuts. A simpler approach is to calculate separatedloxicoherent systems for the positive and negative components of thereal and imaginary parts, although in some cases these can be combined.FIG. 34E shows, analogously to the last line of FIG. 34A, that inregions along the quasi-dominant diagonal where the sign of TCC^((r)) iseverywhere negative, we can set the peak of the T function to −1, andcan reverse the sign of TCC^((r)) when taking square roots in FIG. 34D.This approach can be taken with either the real or imaginary part ofTCC^((r)), in cases where complex-valued kernels are not being employed.Since this amounts to a simple absorption of the sign by {tilde over(T)} and T″, we will essentially follow the same procedure whetherdealing with the complex TCC^((r)) or its real or imaginary part, orwith the positive or negative TCC^((r)) sections of these parts, thoughthe distinctive symmetry of the imaginary part is preferably taken intoaccount, as will be discussed. For simplicity the following equationswill not explicitly distinguish the case of negative-valued TCC^((r)),where it is understood that the sign of the overall contribution to theloxicoherent system will be reversed.

Under all of these sign variants, the local TCC₊ ^((r))′ functions whosegeometric mean is calculated in the second line of FIG. 34D can next beconverted back to the global TCC^((r)) function by applying the equationof FIG. 34B, which results in the third line of FIG. 34D. The FIG. 34Crotated system approximation to the quasi-dominant TCC^((r)) now takesthe form shown in the first line of FIG. 34F. Converting the overallexpression back to the global TCC^((r)) function on the left and toglobal frequency coordinates on the right (by applying the equations ofFIGS. 34B and 34A, respectively) we arrive at the second line of FIG.34F. This already qualifies as a loxicoherent system (but has aremaining deficiency to be discussed shortly), so following the logicintroduced earlier in connection with FIG. 20B, we substitute a T″kernel for the T kernel of the rotated system, as a notational change toindicate that the function can be refined for better matching toTCC^((r)) when used in a loxicoherent system instead of a rotatedsystem.

Since TCC^((r)) is Hermitian, the quasi-dominant diagonal at Δf=+Δf₀will be complemented by an equally strong diagonal at Δf=−Δf₀.Analogously to FIG. 34A, FIG. 34G defines appropriate local frequencycoordinates f_(1,−)′ and f_(2,−)′, whose values will be numericallyclose in regions of significant TCC^((r)) near this secondquasi-dominant diagonal. Through steps similar to FIGS. 34B-34F we thenarrive at FIG. 34A, which represents the portion of an addedloxicoherent system that applies near the Δf=−Δf₀ quasi-dominantdiagonal, just as the last line of FIG. 34F applies near the Δf=+Δf₀quasi-dominant diagonal. From Hermitian symmetry it follows that the T″functions for these two regions share a common mirrored shape, as shownin the first line of FIG. 34I, where a common shape function T″ has beenintroduced. Since the +Δf₀ and −Δf₀ portions of TCC^((r)) are fittedseparately in the new loxicoherent system, T″ should depress to 0 anycontributions from the excluded diagonal, i.e. from the distant diagonalon the opposite side of the origin from the peak. In the convention usedhere T″ is defined as a (generally narrow) function that is centered atthe origin; T″ is then given an argument when applied in theloxicoherent system that shifts it to ±Δf₀, as shown in the first lineof FIG. 34I. This means that the required truncation of contributionsfrom the excluded diagonal is effected by defining T″ to drop to 0 whenits argument is less than −Δf₀, as shown in the second line of FIG. 34I.These steps allow the last line of FIG. 34F to be combined with FIG. 34Hto form the loxicoherent system that is shown in the first line of FIG.34J, with subsequent lines defining the terms involved. The FIG. 34Jloxicoherent system is seen to have a slightly different form from thesystems of FIG. 34H or 34I. However, as in other embodiments, each FIG.34J system uses multiple distinct kernels (unlike prior art coherentsystems, each of which is represented computationally by a bilinearproduct of a single kernel), with the FIG. 34J system using the threedistinct kernels T_(a)′, T_(b)′, and T″ (though as noted, T_(a)′ andT_(b)′ are derived as two distinct kernels from a common function T).The loxicoherent form shown in the first line in FIG. 34J gainsefficiency by combining the systems of FIGS. 34F and 34H, and so removesa deficiency in these latter systems that was referred to earlier.

It should also be noted that even though FIG. 34J calculates the T′kernels in terms of a single quasi-dominant difference frequency Δf₀,these expressions may readily be averaged over a range of Δf₀ values;for example (in the case of 2D patterns), over a range of 2D differencefrequencies having the same Euclidean magnitude.

To complete construction of the FIG. 34J loxicoherent system it isnecessary to explicitly calculate the constituent incoherent systemkernel T″. This can be accomplished using FIG. 34K, which may be derivedusing the same logic as was used to derive FIG. 22J (also making use ofthe last line of FIG. 34I). The equivalent of 3 FFT-based convolutionsare required to apply the FIG. 34J loxicoherent system during imagecalculations, as shown in FIG. 34L; for example, the spatial domainquantities m′_(a) and m′_(b) may each be calculated using what isessentially a convolution, and then a further convolution of theirproduct with the re-phased t″ kernel may be carried out to obtain theintensity contribution ΔI^((r))(x). (The FIG. 33 kernels only entail 2convolutions per system, and are typically a more appropriate choicewhen Δf₀ is small.) FIG. 34L has a compact structure that is convenientfor computation, but its relationship to the simpler lth loxicoherentsystem term in the more basic FIG. 20H decomposition can be seen byre-writing FIG. 34L as the expression shown in FIG. 34M, which isreadily derived from FIG. 34L when T (defined in FIG. 34J) isreal-valued (with t denoting the Fourier transform of T). FIG. 34M showsexplicitly how loxicoherent kernels in the unprimed form can bere-phased in order to map properly to TCC^((r)) content at nonzero Δf₀.In physical terms, the FIG. 34J loxicoherent system includes twoconstituent coherent systems which are differently displaced in thefrequency domain, though their apertures have a common local shape, withthe transmitted amplitudes from these constituent coherent systems beinginterfered with one another after having the two different tilt phases±Δf₀/2 prismatically removed, with this interference pattern then beingused as an input to a constituent incoherent system whose output isupshifted by Δf₀.

A specialized form of the FIG. 34J loxicoherent system will now bediscussed that is particularly well-suited for correcting the imaginarypart of TCC^((r)) using real-valued kernels. This specialized form mayalso employ a different symmetry in the T″ kernel from that expressed inFIG. 34I. As has been discussed in connection with FIG. 15F,Im[TCC^((r))] tends to have a more complicated symmetry thanRe[TCC^((r))], in that the imaginary part of the near-DC fin is actuallyzero-valued where the difference frequency Δf is exactly zero, though itnonetheless becomes relatively large (in magnitude) at finite differencefrequencies that are close to 0, exhibiting a ridge structure that isantisymmetrically split between positive and negative ridges (referringonly to the imaginary part of TCC^((r))). In lithographic applicationsthe real part of TCC^((r)) is typically of greater significance than theimaginary part of TCC^((r)), particularly prior to extraction of thefirst loxicoherent system; nonetheless, defocus can have anon-negligible impact via Im[TCC^((r))], even though the impact ofdefocus through the real part of TCC^((r)) (this latter being alsochanged- and generally increased-by defocus) is usually the strongereffect. (Note too that the peak Δf=0 values of the dominant fin are purereal, per FIG. 22E.) Defocus maintains the bilateral symmetries thatlithographic sources are usually designed with. Any asymmetries in thesource shape as physically rendered are usually small in lithographicapplications, as are residual asymmetric lens aberrations, and theseasymmetries may often be neglected when considering high-ordercorrection terms, like loxicoherent systems beyond that for the dominantΔf=0 fin.

Referring to FIG. 35, when correcting the imaginary part of the residualTCC error it is appropriate to use certain specialized variants of FIG.34J, such as those shown in FIG. 35A, 35B, or 35C; each of thesespecialized forms being shown more specifically as the first line ofFIG. 35A, 35B, or 35C. It should be noted that in the FIG. 35B form T″is made antisymmetric (i.e. an odd function, as shown in the last lineof FIG. 35C), while in the other cases T″ is symmetric (even). The T′functions in these loxicoherent systems are determined from TCC^((r)) inone quadrant of the doubly-dimensioned Hopkins domain (e.g. the quadrantwhere f₁>0 and f₂>0), for example by using one of the embodimentsdiscussed in connection with FIGS. 33 and 34. The odd/even symmetriesprescribed in FIG. 35A, 35B, or 35C then act to provide the overallfitted approximation to Im[TCC^((r))] with the proper symmetry in theother quadrants of the domain. The embodiments described in FIG. 33provide a single T′ function, and this function can readily be adaptedto provide a loxicoherent approximation to Im[TCC^((r))] that takes onthe FIG. 35B form, since the FIG. 35B kernels (T′_(a,odd) andT′_(b,even)) can share a common shape T″ within one quadrant of thedoubly-dimensioned Hopkins domain (e.g. where f₁>0 and f₂>0), with thisshape being mirrored with and without a sign change to provide theprovide the appropriate overall even or odd parity required ofT′_(a,odd) and T′_(b,even). Such a procedure is convenient andacceptably accurate in the usual case where Δf₀ is low frequency.

Another approximation that can provide acceptable accuracy may be madewhen extracting loxicoherent systems from a TCC^((r)) that iscomplex-valued, in that it is typically possible to combine theconstituent kernels of two distinct loxicoherent systems that separatelymatch the real and imaginary parts of TCC^((r)) into a single kernelthat can approximately match the full complex-valued TCC^((r)), therebyreducing convolution count. This approximation involves treating themask transmission as real-valued. Most lithographic masks are designedto at least approximate a real-valued transmission, i.e., their generictransmission polarities are nominally real-valued. However, in practicevarious non-idealities come into play. For example, the amplitudetransmittance of mask blank films will vary with propagation angle, andthe transmittance will usually have an imaginary component that, whilequite small, is not entirely negligible. In addition, the finitethickness of practical mask films will generally give rise to scatteringeffects along the edges of patterned apertures, and this scattering istypically modeled as a transmitted edge-field or boundary field that iscomplex-valued. However, even though the total imaginary-partcontribution of these non-ideal components is usually large enough tomatter in the overall image calculation, it is usually sufficientlysmall as to be neglectable where high-order correction terms areconcerned (such as in the contributions of higher-order loxicoherentsystems), since these high-order corrections are already small evenwhere the dominant real-part contribution is concerned.

Thus, when calculating the image contributions made by high-orderloxicoherent systems, it is usually acceptable to neglect the imaginaryparts of e.g. the blank transmission and the edge fields. For similarreasons one can generally neglect the contribution made by residualasymmetric lens aberrations to high-order loxicoherent kernels, so thatTCC^((r)) residuals can be treated as symmetric. If the real andimaginary parts of TCC^((r)) are individually matched using separateloxicoherent systems, symmetry in the residual TCC will cause thefrequency-domain constituent kernels of these systems to be purelysymmetric or anti-symmetric, and separate matching of the real andimaginary parts will make these individual kernels real-valued.Spatial-domain kernels are obtained as the inverse Fourier transforms ofthese symmetric or antisymmetric real-valued functions, and it followsthat the spatial-domain constituent kernels will be either pure-real orpure-imaginary. Moreover, when convolving these spatial-domain kernelswith a mask transmission that is approximated as pure-real, a factor ofi (i.e. the square-root of −1) may arbitrarily be added to a kernel thatis pure-real, or removed from a kernel that is pure-imaginary, so longas this factor of i is properly added back when the convolution iscompleted. In other words, a kernel can be changed from pure-real topure-imaginary and vice-versa, and when the mask transmission ispure-real this choice will be preserved in the convolution output.

It then becomes possible to multiplex together two separate maskconvolutions involving different T′ kernels into a single convolutionthat uses a complex-valued kernel, as long as the imaginary-partcontributions from asymmetric aberrations and mask-blank non-idealitiescan be neglected, as will usually be the case when the T′ kernels areconstituent to higher-order loxicoherent systems. For example, once thedominant DC-fin has been extracted (with this fin being pure-real alongthe peak), the imaginary part of the remaining TCC residual may then beapproximately matched using a higher-order system of, e.g., the FIG. 35Bform, and the real part of TCC^((r)) approximately matched using, e.g.,a higher-order system of the FIG. 33C form. To improve efficiency, themask convolution that is carried out when applying the latter system maythen be absorbed into one of the mask convolutions used in applying theformer system. This may be done by forming a complex-valued T′ kernelwhose imaginary part is e.g. the T′_(a,even) kernel used in matchingIm[TCC^((r))], and whose real part is the T′ kernel used to matchRe[TCC^((r))].

In some cases acceptable accuracy may be maintained if this merging ofconstituent kernels is extended to the T″ kernels. As has beendiscussed, both the real and the imaginary parts of the TCC^((r)) whichremains after extraction of a primary loxicoherent system will tend tobe dominated by low frequency content, i.e. the real part of TCC^((r))and the imaginary part will both generally have sub-dominant peaks atlow Δf₀ frequencies that flank the removed DC-fin. The sub-dominantpeaks in the imaginary part will generally have Δf₀ frequencies that aresomewhat closer to zero (DC) than the frequencies of the sub-dominantpeaks in the real part, since the former function is antisymmetric andthe latter symmetric; however, accuracy may still remain adequate ifboth systems are handled using a single common T″ convolution, therebyfurther reducing convolution count and runtime.

Returning now to the case of distinct (i.e. non-multiplexed) systemkernels, it should be noted that even though the methods of FIGS. 33,34, and 35 only define T′ kernels in terms of their ability to correctTCC^((r)) along a quasi-dominant diagonal Δf₀, these methods also usethe equation of FIG. 34K to define T″ in a way that provides optimizedimprovement throughout the Hopkins doubled domain. Moreover, asdiscussed earlier, T′ can also be refined to provide additionalimprovement at frequency pairs away from the Δf₀ diagonal. This allowsrefinement to be used to correct a limitation of the FIG. 33 methodsthat arises when they are used to provide an initial T′ kernel, namelythat the FIG. 35B kernel is inherently incapable of providing correctionat difference frequencies sufficiently high that |Δf|>2f, in the casewhere T′_(a,odd) and T′_(b,even) are derived from different mirroringsof a common T′ function. However, refinement of T′_(a,odd) andT′_(b,even) over the full domain as independent variables can providethis high frequency correction.

FIG. 36 shows examples of these different modes of correction; inparticular, FIGS. 36A-36C show for comparison three different TCCresiduals that remain (specifically, the imaginary part of the remainingTCC residual) after three different second loxicoherent systems areapplied in the previously described example involving the free-formsource of FIG. 15A and the film stack of FIG. 15C, the three differentloxicoherent systems are obtained using three of the embodimentsdescribed above, as will be explained. In all cases the secondloxicoherent system is extracted from the Im[TCC^((r))] that was shownpreviously in FIG. 15E.

The example second loxicoherent system used for FIG. 36A employs T′constituent kernels obtained with the simple analytical expression ofFIG. 33C, and uses the system structure shown in FIG. 35B. It should benoted that the FIG. 36A plot of the newly reduced Im[TCC^((r))] is (likethe FIG. 15E plot of the original Im[TCC^((r))]) plotted on the samevertical scale as was used in FIG. 15D, with FIG. 15D showing, asdiscussed above, the more consequential real part of the TCC^((r)) thatremains after 24 OCS kernels have been extracted from the TCC of animaging system that employs the FIG. 15A free-form source and the FIG.15C film stack.

Comparison of FIG. 36A to FIG. 15E shows that the newly added secondloxicoherent system has been able to very substantially reduce themagnitude of the imaginary part of TCC^((r)). To obtain T′ for the newsystem |Δf₀| is set to 0.1 (in direction cosine units) when applyingFIG. 33C, since this difference frequency corresponds to the peak of thequasi-dominant ridge in Im[TCC^((r))] that is visible in FIG. 15Eadjacent to the Δf=0 axis. FIG. 15F shows how this quasi-dominant peakis split between positive and negative regions [with the latter beingshown cross-hatched in FIG. 15F]. T″ for this system has been calculatedusing FIG. 34K.

FIG. 36A illustrates that even the simple FIG. 33C solution is able tovery substantially reduce the magnitude of the imaginary part ofTCC^((r)). FIG. 36B next shows the result of a slightly improvedsolution that is obtained by numerically refining the T′ kernel alongthe segment of the quasi-dominant peak at |Δf₀|=0.1 that lies within theleft-hand quadrant (where TCC^((r)) is positive-valued), in order toimprove the fit to the peak within this quadrant, as discussed above.Symmetry is then relied on to obtain a matched improvement in the otherquadrants. Such refinement is quite fast because it is carried out in alow-dimensioned manifold, in this case the manifold where |Δf₀|=0.1(with our choice in this example being more specifically Δf₀=−0.1).

While the FIGS. 34A and 34B residuals are both considerably reducedbelow the initial residual shown in FIG. 15E, close inspection showsthat neither of these reduction methods provide improvement at largevalues of |Δf₀|, though it should be noted that Im[TCC^((r))] is quitesmall to begin with in these high frequency regions. As discussed above,use of T′_(a,odd) and T′_(b,even) kernels that share a common shape T′within one quadrant does not provide reductions in TCC^((r)) at highdifference frequencies where |Δf|>2f. However, if T′_(a,odd) andT′_(b,even) are refined as independent variables to minimizeIm[TCC^((r))] over the full domain, the imaginary part of TCC^((r)) cansuccessfully be reduced across all parts of the doubly-dimensionedspace. The result of such an optimization is shown in FIG. 36C, whererefinement of T′_(a,odd) and T′_(b,even) over the full domain has inaddition further reduced the residual at moderate differencefrequencies, as well providing an improvement at high differencefrequencies where |Δf|>2f.

One may regard the methods of FIGS. 33 and 34 as attacking the problemof extracting higher order loxicoherent systems by generalizing tonon-zero |Δf₀| the FIG. 22 method for extracting the dominant Δf=0loxicoherent system. This broad class of methods is attractive becauseMercer kernels are inherently inefficient at extracting “diagonallyoriented” content within the doubly-dimensioned Hopkins domain, where(as usual when describing the invention) diagonal refers not toorientations that are slanted between the x and y Cartesian coordinatesof the object patterns, but rather to orientations that mix the two setsof mask coordinates that appear in the doubly-dimensioned Hopkinsdomain. The Δf=0 diagonal (“fin”) dominates as a Gibbs-like residual atthe primary slope discontinuity in the TCC, but more generally thepost-OCS residual TCC will tend to contain additional content alongother diagonals that can be efficiently extracted using additionalloxicoherent systems, but that is not well extracted by additional OCSkernels. Loosely speaking, the methods of FIGS. 33 and 34 may beconsidered to extract content along these sub-dominant diagonals usingenhanced versions of the FIG. 22 method. This approach is efficient aslong as TCC^((r)) content remains concentrated in particular diagonals.It is almost always the case that the extraction efficiency of e.g. asecond loxicoherent system which is obtained using a quasi-dominantoff-diagonal will be quite high (with this efficiency being improved bythe FIG. 34K optimization of T″ over all diagonals, and optionally beingfurther improved by refinement of T′ over the full domain). However, theresidual content in TCC^((r)) becomes more evenly distributed as furtherkernels are extracted, reducing the benefit from using the strongestdiagonal ridge in TCC^((r)) as the basis for choosing later T′ kernels.Of course, even a suboptimal T′ kernel will provide some benefit, andsuch a kernel can be improved by refinement. However, the non-convexnature of the loxicoherent extraction problem makes it desirable tobegin any such refinement with as strong a starting design as possible.Moreover, even though only a very small number of loxicoherent kernelsare typically sufficient in practice to reduce TCC^((r)) to a very lowlevel, it is convenient to have a general method that is suitable forextracting an indefinite sequence of loxicoherent kernels. One suchsystematic approach is to extract TCC^((r)) diagonal by diagonal. Morespecifically, such a systematic approach employs the methods of e.g.FIGS. 33, 34, and 35 to extract TCC^((r)) using a sequence ofloxicoherent systems whose T′ kernels are defined using a succession ofquasi-dominant diagonals, and whose T″ kernels are individuallyoptimized to reduce TCC^((r)) throughout the Hopkins domain. However, itcan be advantageous to define higher order loxicoherent systems withoutrestricting the determination of T′ to consideration of a narrowdiagonal portion of the doubled domain.

This can be done by employing a homotopy method that will now bedescribed, in which an oversimplified but readily solvable problem istransformed, in an incremental (and therefore tractable) way, into theloxicoherent extraction problem of interest. In particular, an initialproblem formulation having a valid but inefficient solution is slowlytransformed, in small homotopy steps, into a problem that provides validand efficient loxicoherent kernels as its solution, where the steps aremade sufficiently small as to allow each new problem to be recast into amore tractable form based on knowledge of the solution from the previousproblem. This homotopy method for kernel determination is of particularinterest because it follows the same computational scaling law as doescalculation of an OCS kernel, albeit with a considerably larger butstill acceptable constant factor. (Note that the scaling here refers tothe relatively small task of computing the necessary decompositionsystems; compute time for the main task of applying these systems duringmask generation is proportional to the number of mask convolutionsrequired, and so is strongly reduced by the novel system sets employedby the invention.)

Before explaining the specific steps used in the homotopy method, thegeneral outlines of the homotopy flow will be described, providing anoverview of how the solution progresses from a result with knowncalculability via FIG. 20A, but with limited practicality, to a finalsolution that constitutes an efficient implementation of the desiredloxicoherent structure of FIG. 20I. More specifically, the initialhomotopy problem will be shown below to have as its solution a simpleversion of the right side of FIG. 20A; one in which the

kernel of the first rotated system (for the new TCC^((r)) residual) issimply diagonalized into a large number of eigenfunctions T′. Forreasons discussed previously, the right side of FIG. 20A can make anaccurate match to the first rotated system (which is a reasonablyaccurate fit to TCC^((r))), if one accepts the (extremely) inefficientexpedient of choosing a relatively large number K of eigenfunctions withwhich to decompose

. Successive steps of the homotopy are then undertaken, with each stepbeing indexed by an iterator n, to reduce K to a desired small valuewithout degrading accuracy (and, in fact, almost always improvingaccuracy). The homotopy problem transformation is governed by parametersK_(n) and γ_(n). Before the homotopy begins, n is considered to have thevalue 0, and the initial number of

eigenvectors that are retained (this initial number being denoted K₀,with each of the K_(n) eigenvectors in a subsequent nth iteration beingdenoted T′ due to their status as constituent coherent filter kernels)is preferably chosen large enough that the retained large set of theseT′ kernels is sufficient to decompose

with high accuracy; for example K₀ might be set large enough to includeall eigenelements with relative eigenvalue magnitude larger than 10⁻⁵.Parameter γ₀ is typically initialized to 0 at the beginning of thehomotopy, and evolved during the homotopy to a final value of 1.

Over an appreciable number of steps (denoted n_(final), which might beas many as about 50), parameter K_(n) is reduced to a value that permitsefficient image calculation, and this value should preferably be setequal to R⁽⁺⁾+R⁽⁻⁾ in FIG. 20I. For example, in a preferred embodiment,the final value of K (denoted K_(n) _(Final) ) is set at 2, whichcorresponds to both a first component that predominantly matches toquasi-positive TCC^((r)) content (so that R⁽⁺⁾=1), and a secondcomponent for predominantly negative content (so that R⁽⁻⁾=1), therebyyielding a final loxicoherent system whose application entails threeconvolutions. In the spatial domain such a system has three kernelcomponents, namely a t′₍₊₎ component which may pre-filter forlargely-positive TCC^((r)) content, a t′⁽⁻⁾ component forlargely-negative content, and a t″ intensity kernel for an outerconvolution along the slanted difference-frequency axis. The homotopyprocedure actually solves for the frequency-domain Fourier transforms ofthese kernels, denoted T′₍₊₎, T′⁽⁻⁾, and T″.

In its intermediate steps the homotopy procedure yields intermediateversions of T′₍₊₎ and T′⁽⁻⁾, which for the nth step are denoted T′_(n,1)and T′_(n,2). These kernels are defined during the homotopy as thedominant eigenfunctions of a matrix denoted

, which is referred to as a homotopy matrix. As will be discussed,

is a kind of amalgam of both TCC^((r)) itself, and the

kernel of the dominant rotated system that is extracted from TCC^((r))via FIGS. 12F and 12G. (In an alternative embodiment,

can be interpreted as an amalgamation of

with a more general corrector that yields TCC^((r)).)

Referring to FIG. 37,

in the nth step of the homotopy is specifically given by FIG. 37A. Forcomputational purposes the quantities appearing in this equation may beassumed to be gridded, and the nth version of the

homotopy matrix is denoted

. The quantities T

C_(n) ^((r)) and

_(n) are scaled and adjusted versions of TCC^((r)) and T″, respectively,as will be discussed. A total of K_(n) eigenfunctions of the nth versionof

are retained, with the kth of these eigenfunctions being denotedT′_(n,k). (As will be discussed, K_(n) does not, in a preferredembodiment, denote a sharp dividing line between wholly retained andwholly discarded eigenfunctions; instead, a more gradual filtering ofeigenfunctions beyond K_(n) is preferably employed.) Usually T′_(n,1)and T′_(n,2) will represent the working versions of the final K_(n)=2loxicoherent filter kernels that will be produced as a completedsolution by the homotopy [these being denoted T′₍₊₎ and T′⁽⁻⁾]. Itshould be noted that in early stages T′_(n,1) and T′_(n,2) will oftenhave a form which is radically different from that which they take atthe end of the homotopy. FIG. 37A indicates that the nth version of

is formed using the T′ and

″ kernels from the previous iteration (n−1).

As shown in FIG. 37B, the procedure of FIGS. 12G and 12F may be used toinitialize

as the optimal rotated system kernel along the f-axis (with the symbol

having been introduced earlier for such a purpose). In some embodiments

is not changed from this initial form during the homotopy. However, inother embodiments

may be evolved during the homotopy, and in such cases the nth version of

can be updated from the previous

_(n−1) version of the f-axis kernel. The T′ functions are initialized asthe eigenfunctions of

, and, per the convention used throughout this invention description,the T′_(n,k) are normalized to absorb the square root of theirassociated eigenvalue. This same normalization convention is followedwhen later iterations of T′ are extracted as eigenfunctions of

. The homotopy also evolves a T″ kernel, which represents a successivelyimproved estimate of the optimal incoherent kernel of the loxicoherentsystem being generated by the homotopy, following the same T″ notationas has been used throughout this description of the invention.

Before being introduced into FIG. 37A, the nth version of T″ preferablyundergoes a scaling and mapped sign adjustment, to be explained shortly,which is reflected notationally by adding a breve diacritic to thesymbol used to designate the interim working incoherent kernel, thisrescaled interim kernel thus being denoted

_(n)″.

As a point of nomenclature, it should be noted that FIG. 37B uses asubscript index of 0 to designate the initial version of

, even though

₀ is obtained from the l=1 rotated system in the FIG. 11A notation. Aslightly different indexing convention has been used in previousequations, wherein a subscript 1 on the analogous quantity

₁ (or a superscript, in the case of

⁽¹⁾) is used to indicate that

₁ is a component of the first rotated system. Our change of notationhere to use 0 as a subscript on

₀ in FIG. 37B reflects a convention in which the subscript representsthe homotopy step number. Under this latter convention the first (n=1)iteration of the homotopy is considered to commence when FIG. 37A isfirst applied, with the preceding calculation of

₀ being regarded as an initialization step.

may be updated in later iterations, as indicated by the n subscript in

_(n), but in some embodiments

is kept at

₀ throughout the homotopy. It should also be noted that FIG. 37Binitializes the working estimate of the dose kernel T″ to the {tildeover (T)} component of the first rotated system. As previouslydiscussed, the T″ of a loxicoherent system and the {tilde over (T)} of arotated system are conceptually different (but related) quantities, andthe working version of the former quantity (denoted T″_(n) beforerescaling and mapped sign adjustment) evolves away from {tilde over (T)}as the homotopy proceeds (and n increases). In the context of thishomotopy embodiment, the quantity denoted T″ should be regarded as aworking or interim kernel candidate; one that only evolves to a usefuldose kernel of a loxicoherent system (for which we have used thenotation T″ elsewhere in this description of the invention) at thetermination of the homotopy.

During the nth iteration of the homotopy, the eigenfunctions T_(n)′ of

constitute a working estimate of the filter kernels of the desiredefficient loxicoherent system, though during early iterations the numberK_(n) of retained eigenfunctions will typically be far larger than thedesired final number of kernels R⁽⁺⁾+R⁽⁻⁾. After

is obtained for the nth iteration via FIG. 37A, its eigenfunctionsprovide the T_(n)′ filter kernel estimates for that iteration. The nthiteration then continues with the application of FIG. 37C to obtain thedose kernel estimate T_(n)″; this is the version of T″ that will be usedin FIG. 37A during the (n+1)th iteration. The derivation of the equationof FIG. 37C is closely analogous to that of FIG. 22J.

When the homotopy iterations commence with the n=1 iteration of theequation of FIG. 37A, the value of γ has been initialized to 0, meaningthat the last term in FIG. 37A drops out, since sin(πγ₀/2)=0. Moreover,cos(πγ₀/2)=1, and since the number of retained Mercer terms K₀ in thefirst line of FIG. 37A is set very large, the T′_(0,k) kernels willcollectively be able to entirely exhaust

₀ to a very good approximation, and it then follows that the second termin the first line in FIG. 37A (i.e.

″₀ (f₁−f₂)Σ_(k=1) ^(K) ⁰ T′_(0,k)(f₁)T′*_(0,k) (f₂)) will approximatelycancel the first term in the second line

$\left( {{i.e.\mspace{14mu}{{\overset{︶}{T}}_{0}^{''}\left( {f_{1} - f_{2}} \right)}}{{\overset{︵}{T}}_{0}\left( \frac{f_{1} + f_{2}}{2} \right)}} \right).$As FIG. 37D points out, this effectively means that only the first termin the equation of FIG. 37A is present at the start of the homotopy.Thus, when the homotopy commences,

will simply be equal to the Mercer expansion of

₀, which is essentially equivalent to

₀ itself, since K₀ is large.

In conjunction with the dose kernel T₀″,

thus provides at the beginning of the homotopy a reasonably accurate butnot very efficient set of filter kernels T′ for the next (generallyhigher-order) loxicoherent system, these initial filters being the largeset of T′_(0,k) kernels (the eigenfunctions of

₀, which are K₀ in number) that allow the loxicoherent system toapproximately reproduce the optimal rotated system, though in aninefficient manner. As the homotopy proceeds, the number of retainedfilter kernels K_(n) is steadily reduced to improve efficiency, whilethe presence of the T″ kernel in the loxicoherent system is exploited toretain accuracy in the face of reductions in the K_(n) count, often evenimproving accuracy over that achieved by the first (optimal) rotatedsystem at the start of the homotopy. In particular, FIG. 37A (whoseessential goal is to produce T′ kernels for a higher-order loxicoherentsystem; these kernels being the eigenfunctions of

) is structured so that T″ will provide this accuracy improvement duringeach efficiency-tightening homotopy iteration in much the same way thatT″ was shown in FIGS. 29A-29E to provide the constituent T′ kernel ofthe first loxicoherent system with a strong accuracy and efficiencyadvantage over conventional coherent kernels. In particular, it wasshown in conjunction with FIGS. 29A-29E that the T′ kernel of the firstloxicoherent system can essentially capture all points within thedominant fin of the pure-OCS TCC^((r)), with this fin being locatedwhere the dose kernel T″ attains its peak value of 1. Conventionalcoherent kernels were shown to be very inefficient by comparison, sinceeach single coherent kernel is essentially able to extract only a singlepoint of the fin. It was further shown that if a coherent kernelattempted to instead extract multiple points along the fin, such acoherent kernel would actually introduce an even larger error in itsmatching to TCC^((r)) at points away from the fin. However, the firstloxicoherent system includes a T″ kernel which is typically very smallat points away from the fin. As was discussed, this means that the T″kernel suppresses the poor match to TCC^((r)) that the T′ kernels of thefirst loxicoherent system would otherwise be engendering away from thefin, allowing the first system to essentially capture all of the finwith just a single T′ kernel. This behavior suggests a strategy that isused in constructing the FIG. 37 homotopy to extract efficient higherorder loxicoherent systems, as will now be discussed.

After the first loxicoherent kernel is extracted, TCC^((r)) willtypically no longer have a fin at Δf=0, but it will still haverelatively substantial content along “diagonal” contours of constant Δf.FIG. 37C ensures that the T″ dose kernel will be strongly peaked in suchregions (particularly when the number of loxicoherent systems that havealready been extracted is modest compared to the number ofalready-extracted coherent kernels). This makes it possible for arelatively small final number K_(n) _(Final) of T′ filter kernels tocapture (in conjunction with T″) a significant portion of the remainingcontent along the sub-dominant diagonals of TCC^((r)) where T″ ispeaked. (Note, however, that the efficiency improvement achieved by thefinal output system after completion of each full homotopy procedurewill typically diminish as the count of already-extracted loxicoherentsystems goes up.) The rescaled version

_(n)″ of T″_(n) that appears in FIG. 37A is designed to facilitate thisdesirable behavior; in particular,

_(n)″ is scaled to lie substantially in the range of 0 to 1. This isdone, as shown in FIG. 37E, by reversing the sign of every element ofT″_(n) that has a negative real part (and then normalizing the peak to1). Once

_(n)″ is formed as a revision to T″_(n) in this way, it is necessary tomake the complementary changes in TCC^((r)); in particular the elementsin every Δf diagonal of TCC^((r)) are multiplied by −1 if thecorresponding element of T″_(n) was multiplied by −1 when forming

_(n)″. The resulting revised version of TCC^((r)) is denoted T

C_(n) ^((r)), and its mathematical definition is reiterated in the firsttwo lines of FIG. 37A. By reversing their element signs jointly,

_(n)″ and T

C_(n) ^((r)) are made to maintain a relationship equivalent to thatexisting between T″_(n) and TCC^((r)), even after the former intensitykernel has been rescaled (as

_(n)″) to become a substantially non-negative quantity. This maintainedrelationship is shown in the last line of FIG. 37E. In addition,

_(n)″ is further scaled to have a peak value of magnitude 1, and tocompensate for this the T′ kernels are scaled in the opposite directionso as to maintain a consistent level in the overall triple products thatcomprise the loxicoherent system.

_(n)″ thus represents in FIG. 37A a working version of the intensitykernel T″, except scaled to a range substantially between 0 and 1. Aspreviously discussed, content in TCC^((r)) will tend to be concentratedalong a relatively small set of diagonal Δf contours, and the evolvingloxicoherent system will tend to give

_(n)″ a value near 1 in these large-error regions in order that theconcentrated TCC^((r)) content be optimally extracted (with optimalityprovided by FIG. 37C). The second line of FIG. 37A (which makes a largercontribution where

_(n)″ is large) then helps ensure that the T′ eigenfunctions of

will tend to reproduce this content. In particular, when K_(n) is setslightly lower than K_(n−1) to begin the nth step of the homotopy, therelatively large value attained by

_(n)″ in regions of large residual TCC^((r)) will cause the remaining T′eigenfunctions (K_(n) in number) to reconfigure themselves to betterrepresent this dominant TCC^((r)) content, since the second line of FIG.37A represents a substantial portion of the matrix beingeigendecomposed. (Even though fewer T′ eigenfunctions are retained forthis purpose, it is the most dominant eigenfunctions that are retained,at least in the simplest embodiment.) If

were given by the second line of FIG. 37A alone, such a reconfigurationmight not proceed very far, since a large reconfiguration would usuallyentail large mismatches in regions where this second line is small, asmay be understood from previous discussion which showed that retainedeigenfunctions/Mercer-terms are not well suited to simultaneouslyreproduce regions where TCC^((r)) is large and regions where TCC^((r))is small. (In other words, as has been shown in FIG. 29, a reducednumber of purely Mercer terms simply cannot reproduce TCC^((r)) well inall regions.) However, in these difficult regions

_(n−1)″ will be set close to zero by FIG. 37C, and the first line ofFIG. 37A therefore sets

approximately equal to its own Mercer expansion in those regions, andthis Mercer expansion is by definition close to

itself. More precisely,

becomes equal in these regions to the Mercer expansion of

in the previous homotopy iteration, and since the homotopy only advancesslowly, this lightly-changed content is easily matched in the nextiteration.

will differ from the true residual TCC in those regions, but thatdeparture will not further degrade the accuracy of the fittedloxicoherent system, since T″ will be correspondingly small there aswell. Moreover, since the strong content in the true TCC^((r)) islargely concentrated in regions of the domain where the second line ofFIG. 37A is large, the loxicoherent system can provide a reasonablyefficient content extraction even though these regions of large T″ willtypically be fairly small in area. In view of the role

″ plays in projecting TCC^((r)) into certain portions of the homotopymatrix (i.e., in the second line of FIG. 37A), while screening othermatrix portions from these changes (first line),

″ can be referred to as a screener-projector function.

Early in the homotopy, γ_(n) will be approximately 0, and the last lineof FIG. 37A will therefore be approximately equal to the product of

_(n−1)″ and

_(n−1). In regions of peak T″, the first factor

_(n−1)″ will be approximately 1, giving the product a value close to

_(n−1). The last line of FIG. 37A will therefore act to concentrate thatcontent of

which lies within high-T″ regions into those next-iteration T′ kernelswhich remain as K_(n) is reduced, since the retained kernels are thosewhich are most dominant (with some exceptions yet to be discussed). Inother words, as fewer kernels are retained, FIG. 37A will inherentlycause a greater portion of

in regions where T″ is significant to be concentrated into theseremaining T′ kernels, as will happen when the updated versions of theseT′ kernels are obtained from the next iteration of

. This process is interrupted, however, because in a preferredembodiment, γ is increased to its final value of 1 in significantlyfewer iterations than are used overall to reduce K_(n) to its finalvalue of R⁽⁺⁾+R⁽⁻⁾. For example, one might typically step γ from itsinitial to its final value during the first ⅓rd of the homotopyiterations, i.e. to fully transition γ to 1 by step n=K_(n,final)/3.Once γ reaches 1, FIG. 37A will eventually act to concentrate content ofTCC^((r)) rather than

into a steadily reduced set of T′ kernels (in regions of large T″),since the last line of FIG. 37 becomes equal to

_(n−1)″ T

C_(n) ^((r)) when γ=1.

After a new set of eigenfunctions T_(n+1,k)′ of

are calculated, T″ may then be updated using FIG. 37C. In a preferredembodiment, a normalized version of the updated T″ will next be createdper FIG. 37E, denoted

_(n+1)″. While

_(n+1)″ will be normalized to have a peak magnitude of 1, its value atsecondary peaks will typically decrease after K_(n) is decreased fromits previous value of K_(n−1), if such a decrease was made during theimmediately preceding evaluation of FIG. 37A. It can therefore beadvantageous to execute the next (i.e. n+2) iteration of FIG. 37Awithout decreasing K, since the subsequent iteration of FIG. 37C maythen be able to restore the scope of the large T″ regions. As a rule ofthumb, K_(n) may be decreased in every other iteration of FIG. 37A.

In a preferred embodiment, the reduction in K_(n) that is carried outduring the homotopy is not effected in a direct and literal way, as willnow be explained. At the beginning of the homotopy, K_(n) is typically alarge number. However, as K_(n) becomes a relatively small number, areduction in K_(n) that was effected in the direct way of entirelydeleting one or more of the few remaining T′ eigenfunctions wouldrepresent a substantial jolt to the structure of

, and this jolt may represent a larger change than is desirable in asingle homotopy step. It is therefore preferable to reduce K_(n) by amore sophisticated method than simple reduction of the number ofretained terms in the Mercer series expansion of

. In particular, K_(n) may instead be employed as a parameter of aweighting function, such that the eigenfunctions T′ of

are effectively removed in a continuous way as K_(n) is reduced (i.e. byincreasingly de-weighting them in the Mercer expansion of

, rather than by deleting these eigenfunctions outright. FIG. 37F showshow FIG. 37A may be modified to do this using a weighting function thattakes the form of a shifted half-Gaussian. This may be further modifiedto take into account degeneracies in the eigenvalues.

It is also not necessary that the k=1, 2, . . . K_(n) eigenfunctions T′of

be included in the FIG. 37A or 37E Mercer series terms in the standardordering that is conventionally used for Mercer series (which is toorder the terms by the magnitude of their absorbed eigenvalues).Instead, it may be preferable to order them (at least partially) by theerror with which their associated triple product T″T′T′* matches toTCC^((r)), so that the least valuable eigenfunctions are deleted whenK_(n) is reduced.

Another useful heuristic that may be employed in the homotopy involvesconsideration of the parity of the eigenfunctions of

. With a symmetric lithographic system these eigenfunctions aresymmetric, with roughly half the eigenfunctions being of even symmetryand the other half of odd symmetry. Antisymmetric eigenfunctions have azero at the origin, and this can be disadvantageous in the usual casewhere the lithographic masks of interest produce a strong zero order, aswill be discussed. This problem can be dealt with separately from thehomotopy by employing what we refer to as a DC-monolinear system, to bedescribed below. Alternatively, one may address this issue by inhibitingthe presence of odd-symmetry eigenfunctions in the K_(n,final) kernels.For example, in the common case where K_(n,final) is 2, one may wish toensure that at least one of the final two kernels has even symmetry. Oneheuristic for doing so is to slightly de-weight the odd-symmetryeigenfunctions when reconstructing

following each iterative reduction of K; for example, the amount ofde-weighting might be inversely proportional to the total number ofiterations in the homotopy. One way to choose the constant ofproportionality in such a de-weighting is to set this constant to bejust large enough that the dominant eigenfunction of

has even symmetry when the homotopy commences. Application of such aprocedure may sometimes suggest (e.g. when the dominant eigenfunction isalready even before de-weighting is applied) that the de-weightingheuristic would provide little advantage in a particular case, i.e. thatit could be skipped.

Yet another useful heuristic to employ with the homotopy is to embed thehomotopy in outer loops; e.g., to loop through the homotopy twice. Withsuch a procedure the T′ solution obtained at the end of the first set ofhomotopy iterations is not used as the final solution. Instead, a secondcycle of homotopy loops is undertaken, in which K_(n) is reset fromK_(n,final) back to its initial large value K₀, but where the otherparameters and kernels appearing in FIG. 37A are not reset, but areinstead kept at the values attained at the end of the first cycle ofloops. As previously discussed, each cycle of homotopy loops tends toconcentrate large-TCC^((r)) content into the retained eigenfunctions(while avoiding mismatches in less concentrated regions throughevolution of T″). When the cycles of homotopy are repeated it issometimes possible to “sweep” slightly more TCC^((r)) content into theretained T′ kernels.

It is also possible to recast the homotopy used in the second (orsubsequent) cycle of loops in a form that yields a new modifying factorto improve the output of the first (or previous) cycle, instead ofhaving the new cycle directly evolve the result of the previous cycle.When doing so it can be useful to adopt a more aggressive form for thecorrective homotopy factor whose role is to improve

in regions where T″ has large magnitude (where in its previouslydiscussed form this corrective homotopy factor is the square-bracketedquantity in FIG. 37A). In yet another embodiment the more aggressiveform can also be employed in a first (or only) cycle of homotopy loops,resulting in a modified version of FIG. 37A, to be discussed.

FIG. 37G considers the corrective factor in question more specifically(in the regime where y has reached 1). The first and second lines ofFIG. 37G note that the Sign(T″) factor which appears in the definitionof T

C^((r)) (i.e. in FIG. 37E) as a multiplying factor can also be placed inthe denominator, if we exclude for the moment the case where T″ isexactly 0. If we then consider that the homotopy is structured (per FIG.37A) to propagate this T

^((r)) factor into the retained eigenelements of

(particularly in regions where T″ is large), and further that theloxicoherent system has a structure which multiplies the Mercer productsof the retained eigenelements by T″, we see that if the T

C^((r)) factor is successfully propagated into the retainedeigenelements, the multiplication by T″ will reproduce TCC^((r)) if T

C^((r)) is given the form shown as the last line in FIG. 37G, where thesign of T″ in the denominator is replaced by T″ itself, so that thisdenominator would ideally be cancelled by the multiplying T″ kernel inthe loxicoherent system, if fully propagated into the retained

eigenelements. From this point of view the previous less aggressive formfor T

C^((r)) shown in the first line of FIG. 37G (and used previously in FIG.37E) can be understood as a highly regularized version of the moreaggressively correcting form shown in the last line of FIG. 37G.

To use this more aggressive form of T

C^((r)) in a second (or subsequent) cycle of homotopy loops that yieldan improving factor for the output of a previous cycle of loops, we canset K_(n) _(Final) to 1 during the second cycle, and then multiply theretained eigenelements of the earlier cycle by the single outputeigenelement of the second cycle, thus obtaining improved T′ kernels forthe loxicoherent system as a whole. When setting out the homotopy forsuch an improving factor it is useful to employ the notation shown inFIG. 37H. The first two lines of FIG. 37H introduce the symbol B todenote the Mercer product of the retained eigenelements from theprevious cycles of homotopy; factor B (which is a function of f and Δf)is left unchanged in the set of cycles to follow. (It should be notedthat while B will be described for convenience as the solution from aprevious cycle of homotopy loops, the method to improve B that iscurrently being described can, in general, be used to improve a solutionB obtained by any method.) The homotopy equation for the subsequentcycle of loops to improve B (to be explained shortly) will preferablyinclude the aggressive form of T

C^((r)) shown in the last line of FIG. 37G, and since this aggressiveform can also be used in the first (or only) set of cycles, it is worthnoting that the homotopy equations to follow can be adapted to the casewhere no previous cycles are involved by simply setting B to 1, asexpressed by the 3rd and 4th lines of FIG. 37H. In either case, theinterim loxicoherent system that is formed during each loop of the newhomotopy will be given by the product of B with both the current T″, andthe Mercer product of the retained eigenelements of the current

(i.e., the currently retained eigenelements of the second cycle homotopymatrix). The product of the first two of these factors is denoted C, asexpressed in the last line of FIG. 37H.

FIG. 37I shows the result of replacing the T

C^((r)) term used in FIG. 37A with the aggressive variant given by FIG.37G, and with the latter being further adapted for use in a second (orsubsequent) cycle of homotopy loops by replacing T″ in the denominatorwith C. In addition, to avoid extreme ill-conditioning in cases where Capproaches 0, the reciprocal of C has been passed through a sigmoidfunction which limits the value of the reciprocal to a saturated leveldenoted t (not to be confused with the spatial domain kernels t′ andt″), where t might be set to e.g. 100 times the reciprocal of themaximum value of C. Any of the well-known standard forms of sigmoidfunction may be used, such as the hyperbolic tangent function shown inFIG. 37J.

FIG. 37K shows another homotopy embodiment that is designed to improvethe output of a previous set of homotopy cycles, in which the rescaledT″ used in FIG. 37I to form the screener-projector term (and also usedpreviously as the screener-projector in FIG. 37A, where it was denoted

) is replaced by a similarly rescaled C. This rescaled C can beconsidered to include a factor of C in its numerator that may be used tocancel the factor C which appears in the denominator of T

C^((r)) when the aggressive variant is used. Since the denominator iscanceled there is no need to regularize it with a sigmoid function, andFIG. 37K shows the form that the homotopy matrix then assumes. It willbe clear to those skilled in the art that other functions can usefullyserve as suitable screener-projector functions if rescaled to the range0-to-1; for example, instead of using T″ or C, one can alternatively usethe point-by-point maximum of TCC^((r)) and C, or the point-by-pointsquare root of the summed squares of TCC^((r)) and C.

A useful heuristic during homotopy cycles of the kind underconsideration (i.e., that provide a modification to a previouslyobtained solution B) is to replace (at the end of every loop) eacheigenvector T′ of the homotopy matrix with an optimal linear combinationof the eigenvector and a constant vector (using the same optimalcoefficients for all eigenvectors); this has the benefit of guaranteeingthat the solution improves on the solution B from the previous cycle ofloops (or at least does not degrade it), while at the same timeimproving the eigenvectors of the homotopy matrix as well (thoughusually by only a small amount). In mathematical terms, the set oflinear combinations that serve as new eigenvectors (referred to as aremapping) is made optimal if the coefficients (denoted α and β) of thelinear combination are chosen to minimize the matching error E_(Remap)that is defined in FIG. 37L. In FIG. 37L the kth eigenvalue of

is denoted λ_(k), and the kth eigenvector Ω_(k). Since the allowedremappings include the case where β=0 and α=√{square root over(K/Σλ_(k))}, i.e. a remapping which sets the output of the currenthomotopy to unity (thereby preserving the solution from the previouscycle of loops), and since the allowed remappings further include thecase α=0 and ββ1 which leaves the eigenelements unchanged, the remappingsolution that minimizes E_(Remap) will ensure a solution that is atleast as good as that provided by the direct eigenelements of

, and also at least as good as that from the previous cycle of loops; inpractice this solution will generally be better than the latter andslightly better than the former, at least in the early loops. Instraightforward fashion it can be shown that the optimal α and β aresolutions to the simultaneous equations given in FIG. 37L, whosecoefficients are defined in FIG. 37M. These FIG. 37L equations arecubic, but since there are only two unknowns they can be solved toglobal optimality quite rapidly by standard numerical methods. A furthersimplification can optionally be made to FIG. 37A by treating α and 1−βas small quantities whose high powers can be neglected.

It will be clear to those skilled in the art that the approach describedin FIGS. 37L-37N can further be used to remap the homotopy matrixeigenvectors with other combining vectors besides a neutral unityvector, thereby further improving convergence speed. One can, forexample, replace (in each iteration) the first homotopy eigenvector byan optimal linear combination of that vector with the first eigenvectorof TCC^((r))*Sigmoid[1/C,t], taking into account (with optimalweighting) the matrix composed of the Mercer products of the remaininghomotopy eigenvectors.

The above-described refinement procedures and heuristics can improve thekernels of high-order loxicoherent systems, but the unadorned homotopymethod of FIGS. 37A-37F can provide highly efficient loxicoherentsystems even without such improvements. For example, FIG. 38 shows theresult of extracting a second loxicoherent system for the C-quad testcase, obtained using the basic homotopy method of FIGS. 37A-37F with 40iterations (without adding any of the above refinements). Morespecifically, FIG. 38 shows the residual TCC^((r)) error that remainsafter a homotopy-derived second loxicoherent system is extracted fromthe TCC^((r)) of FIG. 24, which, as has been discussed, itselfrepresents the greatly reduced TCC^((r)) that remains after extractionof a first loxicoherent kernel from the 24-OCS-kernel TCC^((r)) of FIG.14.

It is seen that an appreciable further reduction in TCC^((r)) isobtained in progressing from FIG. 24 to FIG. 38 via the extraction of asecond loxicoherent system. This process may be iterated to extractstill higher order loxicoherent systems. The homotopy solution for eachloxicoherent system may be further refined by the optimization methodsdescribed above, but this has not been done in the example of FIG. 38.

A point to be noted here is that the computational cost of the homotopymethod follows the same scaling law as the eigendecomposition used tofind standard OCS kernels. (The computational cost of eigendecompositionis generally considered to be cubic in the grid-point length of theeigenvectors.) It can be appreciated that the homotopy method doesentail a large relative numerical factor in its cost when compared tosimple OCS decomposition, namely the number of homotopy iterations thatare undertaken (typically of order 20 to 50). However, currentcomputational lithography practice accepts even larger relative factorsin certain computations that exhibit this same scaling, in particularduring OPC preparations when so-called “storm analyses” are carried out,in which large numbers of TCC decompositions are assessed in an effortto find focal-plane and image-plane settings that best match calibrationdata. Such storm analyses do not require inclusion of high orderloxicoherent terms, and the cost of subsequently adding high orderloxicoherent terms to the final TCC decomposition (which makes use ofthe focus and image-plane settings obtained from the storm analysis)will only be moderate in terms of precompute time. (And, of course, thetotal precompute time is quite small compared to the subsequent cost ofcarrying out OPC, and in the latter dominant calculation the high orderloxicoherent systems provided by the homotopy can significantly improvecompute time.)

Discussed now is an additional specialized extension of the loxicoherentkernels to the case of Δf≠0.

While it has been demonstrated that the Δf=0 case is particularlyimportant because of the large slope-discontinuity that arises when thepupils in the Hopkins diagram are fully overlapped, there is, inaddition, a weaker discontinuity that is generally present in verydifferent regions of the Hopkins domain, namely the regions near|Δf|=2NA in direction cosine units (or 2NA/λ in reciprocal pitch units),corresponding to intensity frequencies where the two pupils in theHopkins diagram just become fully separated. Though milder than theprimary discontinuity at Δf=0, this weaker discontinuity may give riseto a non-negligible TCC residual that is well-separated from the DC fin,such as that region outlined as ‘B’ in FIG. 39A, where the plottedresidual TCC is that shown previously in FIG. 14, corresponding to aC-quad source. In FIG. 39A the strong content in the Δf=0 fin is seen tobe supplemented by “horn-shaped” content near |Δf|=2NA=2.7 indirection-cosine units, marked (in its positive portion) as region “B”.Although the term “fin” seems less apt as a descriptor for this highfrequency content than it was for the near-DC fin associated with thefirst loxicoherent system, we may refer in general to any region ofsubstantial TCC^((r)) content that is located away from Δf=0 as a“non-DC fin”, particularly when the content in question is bothdiagonally oriented (i.e. associated with a particular value of |Δf|)and is a consequence of a discontinuity involving the lithographicoptical system, as will be seen to be the case here. In brief, theresidual content in region B arises because a circular pupil has nocorners, so that the overlap area in the Hopkins diagram dropsprecipitously as the separation of f₁ and f₂ reaches the bandlimitedvalue, a behavior that is difficult for smooth OCS kernels to track.This gives rise to an accuracy loss whose behavior and mitigation aredescribed in further detail below.

In most cases this non-DC fin near the band limit is only modestlyreduced by the first loxicoherent system. For example, if one comparesthe post-OCS TCC^((r)) shown in FIG. 39 (and shown previously in FIG.14) to the reduced TCC^((r)) that results after the addition of thefirst loxicoherent system, this latter TCC^((r)) having been shown inFIG. 24, one sees that a substantial portion of the non-DC fin near thebandlimit (where |Δf|≈2.7) remains after the first loxicoherent systemhas been extracted. It may further be observed in the FIG. 24 plot thatthe non-DC fin, though modestly attenuated by the first loxicoherentsystem, has taken on a fairly significant magnitude in a relative sensecompared to the content remaining in TCC^((r)) at other frequency pairs(even though its magnitude is of course small compared to that of theremoved DC fin). Roughly speaking, the non-DC fin is reduced by theprimary loxicoherent system because the T″ constituent incoherent kernelprovides a least-squares optimal reduction in TCC^((r)) over the fullHopkins domain, as has been discussed, but this reduction is locallyless strong than the complete elimination of the DC fin peak that isprovided by the T′ constituent coherent kernel. A more thoroughreduction of the non-DC peak near the band limit can be achieved bydirect application of higher-order loxicoherent systems, as will bediscussed.

The nature of the slope-discontinuity that arises when the differencefrequency reaches the |Δf|=2NA band limit is a consequence of thegeometry that the Hopkins diagram assumes at such frequencies, with thetwo pupils in such a Hopkins diagram being depicted in FIG. 39B. (Forsimplicity FIG. 39B does not show the source.) When the differencefrequency approaches the band limit, i.e. when Δf is just smaller thanthe cutoff at Δf=2NA, the two lens pupils in a Hopkins diagram are seento intersect in a zone which has quite narrow width, since within theoverlap zone the curved rim of each pupil will be almost perpendicularto the axis of separation, due to the circular shape of the pupils.Lithographic sources are generally considered to be rendered with asmall but finite amount of blur, so their intensity profile can beconsidered very smooth on the scale of the sharp lens pupil aperture,even in the case of sources that are normally considered to comprisediscrete poles. As Δf approaches arbitrarily close to the 2NA cutoff, itis therefore reasonable to consider the source intensity to be locallyconstant within the vanishingly small overlap zone, at least as alimiting case. This means that the TCC (if non-zero) will essentially begiven by the relative area of the overlap zone, i.e. the area of theoverlap zone as normalized by the total source area.

From the geometry of the Hopkins diagram, it then readily follows thatas Δf approaches the 2NA cutoff, the TCC will be approximatelyproportional to the 3/2 power of the small quantity 2NA−Δf, as long asthis quantity is positive, i.e., as long as Δf is just within the bandlimit, rather than being just outside it. Referring to FIG. 40, it canmore specifically be shown from simple trigonometry that as 2NA−Δfbecomes small (while remaining positive), the TCC will be given by theupper line of FIG. 40A, where the factor S absorbs the source intensitywithin the overlap zone and the normalization constant, which isessentially the ratio of the total source area to the pupil area. PerFIG. 40B, it then follows that the second derivative of the TCC withrespect to Δf is inversely proportional to the ½ power of (2NA−Δf) asthe band limit is approached. Thus, as with the dominant crease at Δf=0,the TCC will exhibit a slope discontinuity at the Δf=2NA band limit,since the second derivative with respect to Δf becomes infinite at thebandedge, as noted in FIG. 40C. The singularity implicit in FIG. 40B isweaker than that governing the Δf=0 crease (FIG. 5A), and as a resultthe residual TCC error found near Δf=2NA will tend to be small comparedto that present in the dominant Δf=0 fin, as may be seen in the exampleof FIG. 14. However, as FIG. 24 illustrates, the residual error nearΔf=2NA can become significant in a relative sense after the firstloxicoherent system has been extracted.

The TCC^((r)) rise near Δf=2NA can be reduced using higher orderloxicoherent systems, as obtained for example by the homotopy methodpreviously described, or by the methods of FIG. 34. Such a reduction maybe seen in FIG. 38, where the remaining error content near Δf=2NA isseen to be smaller than in FIG. 24, thanks to the extraction of a secondloxicoherent system. The homotopy method used to obtain this secondsystem was not specifically focused on the content near Δf=2NA; fordirected reduction of specific content the method of FIGS. 34J-34M wouldbe more suitable.

Loxicoherent systems have thus far been described which can efficientlycapture portions of the TCC that are recalcitrant to extraction withprior art OCS/Mercer kernels, such as the slope discontinuity near Δf=0that arises from the sharp pupil edge, or the weaker slope discontinuitythat occurs as Δf approaches the band limit, or, more generally, anyToeplitz-like content that is diagonally oriented in the f₁, f₂ space.Beyond this, it can also be advantageous to deploy specialized non-OCSsystems in accordance with the invention that address portions of theTCC whose significance is amplified by the typical character oflithographic patterns, as will now be discussed. In particular, thestrong predominance of the zero (i.e. DC) order in the spatial frequencyspectrum of most IC levels (which has been illustrated for a metal levelin the log-scale plot of FIG. 27A, and which can often become even morepronounced than in the FIG. 27A example when negative-tone processes areused) will be shown with reference to FIG. 41 to amplify thesignificance of TCC^((r)) regions in which one frequency coordinate ofthe doubled domain has magnitude close to zero. Such regions arewell-suited to extraction by loxicoherent systems in general, but onecan also deploy a specialized system according to the invention,referred to as a “DC-monolinear system”, which will extract theseregions even more thoroughly, as will now be explained.

FIG. 41A shows how the frequency-domain Hopkins equation governing theresidual (shown previously in FIG. 20C) can be discretized for use incomputation, using a grid of frequencies that are evenly spaced (with agridstep δf), and with the two frequencies of the doubly-dimensionedHopkins domain being indexed by variables j and k. (FIG. 41A follows acommon convention in using the symbol “N” to denote the number ofgridpoints used in an FFT; it will be clear to those skilled in the artthat this meaning is very different from that of the quantity designated“N” in e.g. FIG. 20I.) As has been discussed, the image contributionfrom TCC^((r)) will generally involve all pairs of sampled maskfrequency harmonics, i.e. all j,k combinations of amplitude pairsM(jδf)M*(kδf). However, other things being equal, the FIG. 41A doublesummation will tend (for most IC levels) to be dominated to a degree bypairs in which j and/or k is 0, since such pairs include the dominantmask zero order. Thus, with many IC levels it is regions TCC^((r))[jδf,0] and TCC^((r))[0, kδf] within the residual TCC that make the largestcontribution to the image (or at least make a very substantialcontribution), with the equivalent regions in the continuous domainbeing TCC^((r))[f₁, 0] and TCC^((r))[0, f₂], as noted in FIG. 41B. Thestrong intensity contribution from these regions is not a consequence ofan inherently large TCC^((r)) magnitude due to poor OCS extraction (asin the case with the Δf=0 fin), but rather it is a consequence of theseregions being strongly sampled (in many cases) by the interfering maskfrequency pairs M(f₁) M*(f₂). For this reason we will refer toTCC^((r))[jδf, 0] and TCC^((r))[0, kδf] as “critical pair” regions (andlikewise TCC^((r))[f₁, 0] and TCC^((r)) [0, f₂] in the continuous case).The single function TCC^((r)) [f, 0] will similarly be referred to asthe “critical axis” function, or simply as the critical axis of theresidual TCC.

Strictly speaking, the continuous TCC^((r)) [f₁, 0] and TCC^((r)) [0,f₂] regions have measure 0 within the doubly-dimensioned Hopkins domain,and their discrete TCC^((r))[jδf, 0] and TCC^((r)) [0, kδf] counterpartsconstitute only a single row and column of the discretized TCC^((r))matrix operator (referring to the 1D pattern case for simplicity; itwill be clear to those skilled in the art that these matrix elements arestacked in a more complicated but still readily referenceable way withinthe TCC matrix operator for 2D patterns). However, when one is employingboth OCS systems and loxicoherent systems to reduce TCC^((r)) over theentire domain, it can be productive to dedicate kernel convolutions tothe full elimination of TCC^((r)) along the critical j=0 row and k=0column (though, as will be discussed, a preferred DC-monolinearembodiment of the invention provides reduction over the entire doubleddomain). Often the M(jδf) M*(kδf) products will fall off substantiallyat even the j=±1 and k=±1 pixels adjacent to the critical row/column,since the integral of even the lowest-order (but non-DC) Fouriertransform kernel over the entire optical ambit will not involve thesteady secular accumulation of amplitude that will typically be presentin the DC integral (i.e. it will instead be oscillatory at a higherinteger harmonic). However, in preferred embodiments the spatial domainconvolutions of e.g. FIG. 1C are calculated over a wider OPC frame thanis covered by the Fourier harmonics used to calculate the TCC (i.e., themask frame is typically somewhat larger than the ambit, as has beendiscussed), and convolutions that are extended in this way can beessentially equivalent to interpolating within the central j=0 row ork=0 column of the frequency domain TCC. It is therefore important totake the finite width of the central row and column into account (thoughat a minimum one need only do so to achieve an appropriate scalingfactor). We will express such width cross-sections using a windowfunction P(f). It will be seen that P(f) constitutes one of the twokernels that are present in each DC-monolinear system. In the simplestcase, P may be a rect function whose value is 1 when |f|<δf/2, and 0otherwise. Alternatively, P(f) may be considered to have the form of theaveraged cross-sectional width exhibited by typical mask content M(f)near the DC peak. Beyond this, the shape of P(f) may, in a moresophisticated embodiment, be instead chosen in a way that providesoptimal TCC^((r)) reduction over the full doubled domain, along withcomplete elimination of the critical row/column. With any of theseembodiments, the shape of the P(f) kernel will be strongly differentfrom that of the other DC-monolinear kernel (which is TCC^((r))[0,f] orits conjugate, as will be shown), allowing the DC-monolinear system tomore precisely target the critical regions of TCC^((r)) than can priorart OCS systems, which are each formed as a bilinear product of twocopies of the same kernel function.

In the simple embodiment where P(f) serves as a narrow window functionthat targets the critical pair region, the contribution that thecritical row/column pair makes to the image intensity is given by thefirst line of FIG. 41C, where the first term in square brackets is thej=0 row contribution, and the second term the k=0 column contribution.In other words, when P(f) is defined to be a simple binary aperturefunction that excludes all portions of the residual TCC outside thecritical pair region, FIG. 41C expresses the contribution that thecritical pair region of TCC^((r)) makes to the FIG. 20C intensityresidual. In the most general case the square-bracketed expression wouldbecome locally incorrect in the central {0,0} pixel where the criticalrow and column intersect. However, we will assume in a preferredembodiment that kernels to extract these critical row/columns aredetermined after at least one loxicoherent system has been extractedfrom the TCC; as a result, TCC^((r))[0, 0] will be exactly 0 since itfalls on the (former) ridge of the extracted DC fin, allowing thecentral row/column intersection region to be neglected. The case ofnonzero TCC^((r))[0, 0] will be considered below. In practice the FIG.41 systems are better behaved numerically when TCC^((r))[0, 0]=0, makingit preferable to employ them in conjunction with a first loxicoherentsystem (of e.g. the FIG. 20B form) to ensure suppression of TCC^((r))[0,0].

When the equation in the first line of FIG. 41C is split into a sum ofseparate double integrals involving the two terms in the squarebracketed expression, the resulting two double integrals are complexconjugates of one another, assuming that all previously extracted OCSand loxicoherent systems have maintained the Hermitian character ofTCC^((r)). The intensity contribution from the critical-pair portion ofTCC^((r)) is then given by the second line of FIG. 41C. If allquantities are Fourier inverse-transformed back to the spatial domain(with spatial domain quantities being represented by lower case symbolsas usual), we arrive at the multiplied pair of mask convolutions shownin the last line of FIG. 41C. In physical terms, the DC-monolinearsystem produces its output by interfering the outputs of two constituentcoherent systems, one with aperture transmission P(f) and the other withaperture transmission TCC^((r)) [f,0]. Since the output interferencemodulation is essentially the product of one portion of the maskspectrum with another portion of the mask spectrum (due to use of twodifferent constituent aperture filters), the output of the DC-monolinearsystem is a nonlinear function of the input mask spectrum, with thisoutput being controlled by two distinct kernel functions. In theserespects the DC-monolinear system resembles the other forms ofloxicoherent system that are employed in the various embodiments of theinvention, but the DC-monolinear system is unique in using twoconstituent systems that are both coherent.

The FIG. 41C expression has the same computational cost as eachloxicoherent system in the basic embodiment of FIG. 20H (the latterbeing used by the primary system that extracts the Δf=0 fin), i.e. thesame cost as each single e term of the second sum in FIG. 20H. However,this cost may be cut by almost a factor of two if one can accept theapproximation that p(x) is approximately constant, which may bemoderately accurate under conditions of e.g. uniform pattern content ina bright background mask, since the frequency domain window P(f) can, inmany such cases, be considered to have a width of only 1 grid pixel.Even when the shape of p(x) is optimized explicitly, one can, forconceptual purposes, often consider the second term in the last line ofFIG. 41C to exhibit only a modest and secondary variation over typicalsimulation fields, for mask types and IC content commonly encountered.Thus, it is usually the first term that provides the most substantivecontribution, and we may loosely regard the entire system as making analmost linear contribution to the intensity through the first termalone, rather than making the usual quadratic (and typically bilinear)contribution. (Of course, strictly speaking the FIG. 41C system is fullyquadratic in m(x), with the second term merely exhibiting lesservariability with many masks.) Since the magnitude of the secondquasi-constant term can be deemed roughly equal to the DC amplitudetransmitted by the simulation frame, we refer to the FIG. 41C system asa “DC-Monolinear System”. In physical terms, the two interferingconstituent coherent systems become equivalent to a holographicrendering of a transmitted amplitude, with a plane wave being used asreference if p(x) is constant.

As a point of terminology, it should be noted that in the spatial domainthe “DC-monolinear system” is considered to include the mask patternfactor m(x), as shown in the last line of FIG. 41C. However, in thefrequency domain one can define the “DC-Monolinear System” to be thesquare bracketed expression in the first line of FIG. 41C, which doesnot include the mask spectrum M(f). This difference is nothing more thanan arbitrary choice of nomenclature.

In a preferred embodiment, P(f) is not merely specified as a windowfunction; instead it is determined as an optimal kernel for theDC-monolinear system, following a similar approach to that used for theprimary loxicoherent system (in the preferred embodiment discussed inconnection with FIG. 22). In the FIG. 22 embodiment, one of the twokernels in the primary loxicoherent system (namely T′) is preferablychosen to exactly match the dominant Δf=0 fin, with the T″ kernel thenbeing chosen to optimally match TCC^((r)) throughout the Hopkins domain.Similarly, in a preferred embodiment, the DC-monolinear system exactlymatches the critical row/column of TCC^((r)) by using the functionTCC^((r))[f,0] as one kernel, after which a second P(f) kernel maypreferably be chosen to optimally match TCC^((r)) across the rest of thedoubled domain, by using a procedure analogous to that carried out inFIGS. 22H-22J to obtain T″. To accomplish this, the first line of FIG.41D shows an error metric E_(DC-mono) that should be minimized tooptimally match the DC-monolinear system to TCC^((r)) (the DC-monolinearsystem being the quantity shown in square brackets), where now P(f) isallowed to be complex-valued. The criterion for minimizing E_(DC-mono)is that if one introduces a first-order variation e in the value of P(f)at an arbitrary location f_(per), in the frequency domain, there shouldbe no first-order change in E_(DC-mono) if P(f) has the optimal shape, arequirement expressed in the second line of FIG. 41D.

Following straightforward algebraic manipulations, we then arrive at thesolution shown in FIG. 41E. In a preferred embodiment the DC-monolinearsystem is applied after the primary loxicoherent system, in which caseTCC^((r))[0, 0]=0 when FIG. 41E is used. It follows from FIG. 41E thatP(0)=1 in this case, and the DC-monolinear system shown in squarebrackets in the first line of FIG. 41D (whose spatial domainequivalent-with included mask content m(x)—is shown as the last line ofFIG. 41C) will then exactly match the critical row/column of TCC^((r)),and moreover will simultaneously provide a least-squares optimalreduction in TCC^((r)) throughout the Hopkins domain.

As with the simpler embodiment in which the p(x) kernel of theDC-monolinear system contributes a uniform plane-wave reference to anoutput holographic interference pattern, the preferred embodiments ofthe invention that are based on a least-squares optimal p(x) willlikewise produce a holographic interference pattern as output. In bothcases this holographic output is an interference rendering of the maskamplitude that is transmitted through a constituent coherent system (inpreferred embodiments, a first constituent coherent system), wherein thepupil aperture transmission of this first constituent system is thecritical axis function TCC^((r))[f, 0]. However, in preferredembodiments that use an optimal p(x), the reference beam is itself acoherently transmitted mask amplitude, namely a mask amplitude that isimaged through a second constituent coherent system, in particular asecond constituent coherent system having aperture transmission P(f). Inphysical holographic systems the intensities of the direct object beamand the direct reference beam are often separated from the holographicinterference pattern using simple angular divergence, but to carry outdimensional compensation in masks it is only necessary to determine theholographic rendering computationally, and thus the DC-monolinear systemcan simply be defined as the interference of the object and referenceamplitudes after they are transmitted from the two constituent coherentsystems, as in the last line of FIG. 41C. Moreover, since both of theseinterfering waves will typically have a complex and deeply structuredform, there is little point in identifying one wave as the reference andthe other as the object; FIG. 41C can more straightforwardly bedescribed as an interference between the mask amplitudes transmitted bytwo constituent coherent systems, a description that applies even in thespecial case of constant p(x), if one of the constituent systemapertures passes only the zero order.

The advantage provided by the FIG. 41C system in exactly matching thecritical-pair region of TCC^((r)) when TCC^((r))[0, 0]=0 is magnified bythe fact that a Mercer term [i.e. an OCS-like term of the formΨ(f₁)Ψ*(f₂)] will usually have considerable difficulty extracting thisregion when TCC^((r))[0, 0]≅0. This difficulty arises because one or theother Ψ kernels in the OCS system will drive the product almost to zeroin these regions, as shown in FIG. 41F. Note that TCC^((r))[0, 0] willin fact be exactly zero by construction after a first loxicoherentsystem has been extracted, and in a preferred embodiment theDC-monolinear system is used in conjunction with a primary loxicoherentsystem. Even in the alternative case when all previously extractedsystems have been OCS, it is often true that TCC^((r))[0, 0] (which isthe value of TCC^((r)) at the center of the Δf=0 fin) will be smallerthan at most other frequencies along the fin. However, in situationswhere only OCS systems have been extracted, it may still be the casethat TCC^((r))[0, 0], though small compared to other TCC^((r)) valuesalong the fin, will nonetheless have appreciable magnitude compared tothe value of TCC^((r)) at points away from the fin, including points inthe critical-pair j,k=0 row/column. In such cases one can, as analternative, supplement the already-extracted OCS kernels with theessentially Mercer kernel shown in FIG. 41G. The FIG. 41G kernel(denoted

in the spatial domain and

in the frequency domain) will fully extract the critical-pair row/columnof TCC^((r)), as does FIG. 41C. However, the FIG. 41C system provides amore accurate matching than the FIG. 41G system, since it does notintroduce erroneous TCC content outside the critical row/column. (Infact, FIG. 41C will improve the fit to TCC^((r)) in these regions viaFIG. 41E, thanks to its use of two kernels [TCC^((r))(0,f) and P(f)]which are strongly distinct from one another.) Nonetheless, FIG. 41Gmight expediently be employed in an OCS-only extraction strategy, sinceFIG. 41G is relatively easy to implement in prior art OPC codes thatonly support OCS kernels.

A related side point may be made in this context, namely that theinvention permits an embodiment as yet unmentioned which, while notparticularly accurate, is (like FIG. 41G) relatively easy to implementin prior art OPC codes. In particular, in one embodiment of theinvention a purely incoherent system is fit to the TCC^((r)) thatremains after the coherent system set has been extracted. Technicallyspeaking, such an incoherent system qualifies as a limiting-caseloxicoherent system, specifically a loxicoherent system in which theconstituent coherent system is given such a large and open aperture asto essentially pass on an unaltered copy of the transmitted maskintensity to the constituent incoherent system. However, forcomputational purposes the constituent coherent system may then beomitted since it essentially acts as a null system, and therefore thisembodiment may be implemented as a somewhat modest modification to anOCS code, namely a modification which allows the kernel convolution tooperate on a squared amplitude transmission rather than an amplitudetransmission.

A further point to be noted here is that the FIGS. 41C and 41G kernelscan readily be applied to the 4D TCC^((r)) that governs the imaging of2D patterns, even though as written these equations follow the practice(employed generally in our description of the invention) of onlydisplaying a 1D pattern coordinate x for simplicity.

With a partial exception to be discussed, the same comment about 2Dsuitability applies to the other novel TCC decomposition systems of theinvention. For the most part, 2D loxicoherent correction of the 4Dresidual TCC error can be carried out using the same procedures thathave, for the sake of simplicity, been described above using nominal 1Dnotation. Generalizing the various kernels used by these procedures(such as the T′ and T″ functions) from 1D to 2D is largelystraightforward, and should be considered implicit in the aboveequations. In 2D the arguments of these functions (such as f₁, f₂, Δf,or f) become two-element vectors, i.e. these spatial frequency argumentshave x and y components in the 2D case, so that T′ and T″ becomefunctions with 2D domains, and TCC^((r)) becomes a function with 4Ddomain, with an x and a y dimension being present in each relevantcomponent or sub-manifold of the doubly dimensioned Hopkins domain. Thearguments of the spatial-domain functions ψ, t′ and t″ that appear ine.g. FIGS. 22H and 22I likewise have x and y components in the 2D case.Ordinarily the associated x and y coordinate axes would be chosen asthose of the x and y design coordinates with which IC patterns arecustomarily laid out. The equations and formulas used to obtain thesekernels can in general be immediately extended to 2D by using the 2Dversions of the various operations involved. However, the fundamentalphysical mechanism underlying the slope-discontinuity in the TCC thatdrives TCC^((r)) does include a complicating behavior with 2D patternsthat is not seen in 1D. As will be discussed, this complicating behaviorcauses the fin cross-section (which is characterized by the T″ function)to have a potential azimuthal dependence at frequencies away from thefin peak that may not be captured as rapidly with a single kernel as ispossible with 1D patterns, even after T″ is made a 2D function (i.e. afunction of Δf_(f) and Δf_(y)).

Compensation of this off-peak azimuthal dependence can be effected in adirect way by dedicating separate kernels to different azimuthal zones,as will be explained.

It should be noted that the slope discontinuity and resulting finstructure in TCC^((r)) arise for the same reasons with 2D patterns ashave been illustrated above using 1D examples for simplicity. When theseexamples are extended to 2D, the fin structure can be observed in thefull four dimensions of the TCC (though the 4D fin-like errorpredominance is difficult to convey graphically), indicating that theloxicoherent correction can usefully be applied with 2D patterns for thesame reasons as with 1D patterns. It may be seen that most terms on theright side of the second line of FIG. 5A are independent of theorientation chosen as “x” (though in the context of IC mask design “x”customarily denotes the axis designated as horizontal in the IC designlayout), the one exception being the mildly-varying polar cosine factor|cos θ_(f)|. This means that the “crease” discontinuity in the TCC willgenerally have substantial magnitude independent of the x,y orientationchosen for Δf in the second derivative. While FIG. 5A is only exact inthe idealized regime of aberration-free scalar imaging, a genericGibbs-like fin or spike of error will nonetheless be generally foundacross all azimuths of a realistically calculated 4D residual TCC. Thisfin structure is strongly peaked across two dimensions because of theexpanded TCC dimensionality, i.e. the error spike takes the form of ahigher-dimensioned fin whose maximal ridge (i.e. peak) is a fully 2Dlocus that spans both f _(x) and f _(y) in the case of general 2Dpatterns, and this fin will typically be narrow in cross-sectionthroughout the 2D neighborhood spanned by Δf_(f) and Δf_(y) thatsurrounds each point along the 2D peak of the fin. (Here we continue touse terms like “ridge” and “fin” that appropriately conveyed thecharacter of these shapes in the previously shown 2D TCC examples, eventhough such simple descriptors are not literally applicable to the morecomplex 4D versions.)

Considering first the extended shape and dimensionality of the“ridgeline” or peak of the fin when patterns are 2D, FIG. 42A shows adensity plot of the 2D fin peak TCC^((r))(f,f) in the case of the C-quadsource example previously discussed, with the plotted residual errorbeing the peak in TCC^((r)) that remains after 24 conventional OCSsystems have been extracted. Here f is a two-dimensional spatialfrequency that has x and y components, with these Cartesian componentsserving as the plot axes in FIG. 42A. The spatial frequency componentsare given in direction cosine units (including a multiplying factorequal to the coupling index of 1.44), and the right and upper plot axesalso show the interference pitch P (in nanometers) associated with eachdirection cosine, i.e. P≡λ/f. Note that even though TCC^((r)) isconsiderably smaller in some regions of FIG. 42A than in others, everypoint in the plot that falls within the system bandpass is a point onthe 2D peak (i.e. maximal “ridge”) of TCC^((r)) within the overall 4Ddomain of the TCC, with this “ridgeline” being the two dimensional locusplotted in the figure, so that TCC^((r)) falls off rapidly in the unseen2D manifold perpendicular to the two plotted f dimensions (this unseenmanifold being the {Δf_(x),Δf_(y)} manifold). The 2D space of FIG. 42Athus represents the full two-dimensional extent of the more limited 1Dridgeline along the fin peak that was shown dashed in FIG. 14. Thedashed trace along the FIG. 14 1D fin peak is in fact a cutline tracealong the x axis of FIG. 42A.

The full TCC at each of the frequency pairs whose post-OCS TCC^((r)) isplotted (i.e. depicted as a graylevel density) in FIG. 42A can becalculated from a Hopkins construction in which the two pupil aperturesare made coincident, in accordance with the fin-peak condition that Δfequals 0 in both its x and y components. While each such point isgenerally a local peak or spike of TCC^((r)) within the unseen 2D Δfmanifold orthogonal to the plotted point, it is clear from FIG. 42A thatthe height of this 2D fin peak varies considerably over the 2D fmanifold. In particular, the locations of largest error (which in 1Dwould be “summits” along the “ridgeline”) are seen to take the form of 4pairs of double-concentric near-circular rings (in this non-limitingexample). FIG. 42B shows that these double-concentric near-circles arecentered on the four poles of the C-quad source, whose locations areindicated in FIG. 42B using dashed lines, with the centered lensaperture also being inserted as a black circle. Since Δf equals 0 inboth its x and y components at every point in FIGS. 42A and 42B, eachplotted point is associated with a Hopkins diagram in which the twopupil circles are coincident. Moreover, since the source position in aHopkins construction does not change as the pupil offsets (i.e.frequencies) are changed, the source that is shown in FIG. 42B can beconsidered to be the source in the Hopkins diagram that is associatedwith each plotted point, with the two coincident pupil circles of eachdiagram being centered on the plotted point in question. At f _(x),f_(y) spatial frequencies in which the coincident pupil edges happen tointersect an edge of one of the four source poles, the rapid variationin the TCC becomes particularly large, and therefore tends to beparticularly difficult for the OCS systems to match. Note that eventhough source blur generally leaves the pole edges much less sharp thanthe lens aperture, it is often the case that source shapes containregions of very rapid intensity variation, e.g. pole edges, albeitblurred, and when these edge regions are roughly coincident with theOCS-problematic sharp pupil aperture in the Hopkins construction, thefin peak becomes particularly pronounced. Qualitatively speaking, thiseffect explains the general character of the T′ filter function, causingcertain f frequencies along the 2D fin peak to contribute more toTCC^((r)) than do other frequencies.

The locus of such particularly problematic f points, e.g. points whereTCC^((r)) is pronounced due to coincidence of the pupil edges with anyone of the source pole edges or corners in, e.g., FIG. 42, will take onthe appearance of nested generalized epicycloid-like curves, where eachsuch curve in a nested pair takes on the rough appearance of a circle ifthe associated source pole is small, and/or roughly circular. We mayenvision each epicycloid-like curve as being formed by tracing thecenter of the pupil circle (actually the pair of aligned pupil circles)as the pupil circle is “rolled” around the source pole perimeter (i.e.,with the pupil circle being “in contact” with the source poleperimeter). In the case of the innermost curve of the nested pair, weshould envision this “rolling” construction as being carried out withthe bulk of the source pole positioned inside the aligned pupil circles.

FIG. 42C shows one Hopkins diagram from such a rolling sweep, namely theHopkins diagram for the {f _(x),f _(y)} frequency pair that is labeled U(with Δf_(f)=Δf_(y)=0). The diagram is shown superimposed on the sameplot of the 2D fin peak that appears in FIGS. 42A and 42B. As discussed,this 2D peak in the TCC error arises from the behavior of the Hopkinsimaging configuration when the two interfering frequencies f₁ and f₂become equal (i.e. to frequency U in the FIG. 37C example). Thering-like “summit” regions of the peak where the TCC^((r)) error isparticularly large can be seen in the figure to arise from thedifficulty that the truncated OCS/Mercer series has in capturing therapid transition that occurs when source points at the steep edge of asource pole diffract the frequency in question (e.g. U) to the sharpedge of the pupil (with example point U being chosen somewhatarbitrarily as the frequency on one such ring that happens to havemaximum f _(y) value). Each inner ring constitutes half the locus offrequencies in this category that are associated with one of the poles,where the inner ring can essentially be constructed by “rolling” thepupil circle around the pole, in the manner suggested by FIG. 42C, withthe ring being traced out by the center of the pupil circle.

Similarly, the outer epicycloid-like curve is essentially formed bytracing the center of the pupil circle as the pupil circle is rotatedaround the source pole perimeter with the source pole outside the pupilcircle, i.e. each pronounced outer circle “summit” within the 2DTCC^((r)) ridgeline peak is formed by the trace of the pupil centerwhile the pupil circle is “rolled” in wheel-like fashion around theoutside of the pole. TCC^((r)) tends to be particularly large for pointsat which multiple epicycloid traces intersect, i.e. where multiplesource pole edges intersect the pupil aperture in this way.

The same sort of epicycloid-like processes are found to govern the shapeof the fin peak with other source shapes. Though not quantitativelyprecise, this construction provides a quick rough determination of thespatial frequencies that will exhibit problematic accuracy in an OCScalculation.

Since FIG. 42A is a plot of TCC^((r))(f,f) for a 2D spatial frequencyf={f _(x), f _(y)}, the previous discussion in connection with FIG. 22Gshows that the square root of this plotted quantity will provide anoptimal T′(f). T′ as calculated in this way will exhibit all thestrength seen previously in the 1D examples provided above, andessentially the only impact from considering 2D spatial frequencies isthat T′ becomes a numerical function of two arguments, f _(x) and f_(y). One can similarly apply FIG. 22J to calculate the incoherentkernel T″(Δf), and, per the earlier discussion of this kernel and therelated rotated system kernel

(e.g. in connection with FIG. 17B), such a procedure essentiallydetermines T″ as a mean cross-section of the fin. However, the fact thatthe integrals in FIG. 22J are merging content across a 2D subspace meansthat the reduction of the 2D fin's varying 2D cross-section to a singlefunction may average over a greater share of the off-peak peripheralstructure present in TCC^((r)) than occurs with a 1D slice of the fin(since each T″ off-peak value is a 2D average rather than a 1D average).Nonetheless, FIG. 22J will continue to extract the critical central finitself; this is a key benefit, since, in 2D as well as 1D, this centralfin is the dominant error component in TCC^((r)) before the firstloxicoherent system has been extracted. It is in the weak fine structureaway from the fin that the convergence provided by a single T″ kernelcan be less complete in 2D than 1D, an effect whose mitigation will nowbe discussed.

The weak off-peak fine structure in T″ and the averaging behavior ofFIG. 22J may conveniently be investigated by first choosing 0 as thevalue of parameter p in FIG. 22J, and further by making the explanatoryapproximations in the denominator that

${{{T_{1}^{\prime}\left\lbrack {\overset{\_}{f} + \frac{\Delta\; f}{2}} \right\rbrack}{T_{1}^{\prime*}\left\lbrack {\overset{\_}{f} - \frac{\Delta\; f}{2}} \right\rbrack}}} \cong {{T_{1}^{\prime}\left\lbrack \overset{\_}{f} \right\rbrack}}^{2}$and${{D\left\lbrack {\overset{\_}{f} + \frac{\Delta\; f}{2}} \right\rbrack}{D\left\lbrack {\overset{\_}{f} - \frac{\Delta\; f}{2}} \right\rbrack}} \cong {{D\left\lbrack \overset{\_}{f} \right\rbrack}^{2}.}$Both approximations are reasonable in the critical region near the fin.Given these investigational simplifications, FIG. 22J may be understoodas simply calculating the T″ kernel for each value of Δf as an averageof TCC^((r)) (as windowed by D) over all in-window f values along thecontour parallel to the f axis at which Δf takes on the specified value,where it is understood that with 2D patterns the f “axis” actuallyrefers to the 2D sub-manifold spanned by axes {f _(x),f _(y)}. In otherwords, T″ is simply an averaged 2D cross-section of TCC^((r)) in thedifference-frequency directions, with this average being made over all{f _(x),f _(y)} frequencies within window D. The cross-sectional “axis”Δf similarly refers to the 2D sub-manifold {Δf_(f), Δf_(y)} when 2Dpatterns are considered, and the value of T″ at each 2D differencefrequency {Δf_(x), Δf_(y)} will then approximately be an average of the4D TCC^((r)) function over all possible {f _(x),f _(y)} coordinates.

FIG. 43 shows three plots of such f-averaged TCC^((r)) cross-sections,namely FIGS. 43A, 43B, and 43C, with three different options having beenused for window D in the three plots. Each plot can be interpreted asshowing a different determination of the T″ kernel, in each caseplotting the T″ kernel or averaged cross-section over the full 2D Δfsub-manifold. The three plots will be used to explain how azimuthalvariation in the cross-section can largely be eliminated. In each of thethree cases both a density plot and a surface plot of the averagedcross-section are shown for clarity.

As noted above, an averaged cross-section like those in FIG. 43 may beused to calculate T″ with reasonable accuracy, but for ordinarycomputational purposes one would need to use a D window in doing so thatspans a reasonable portion of the TCC^((r)) content. However, in FIG.43A we have chosen for explanatory reasons to use a window D that (quiteunusually) only includes a single point; in particular, FIG. 43A showsthe cross-section of TCC^((r)) at the f location of maximum error, whichin this example occurs at a spatial frequency of about 0.7 in directioncosine units. (This position of maximum error is located along the f_(x) axis.) Because of the point-like D window employed, there isactually no averaging in the FIG. 43A cross-section; in this extremeexample the plotted analogue of T″ is simply the cross-section of thefin peak at its highest point (where “analogue” has been added as aqualifier to reflect the fact that a very unusual choice of D has beenmade). This T″ cross-sectional plot shows a strong central spikecorresponding to the dominant fin, and one finds that the fin's centralspike is successfully reproduced in this way (i.e. taking the form of anarrow spike at the center of a 2D Δf cross-section) with virtually anychoice of D. The FIG. 43A cross-section also shows a weak fine structureaway from the fin, and in this example the fine structure is seen tocomprise faint “spokes” or “arms” oriented at roughly 45° in the x,ydomain, along with some vaguely ring-like peripheral content. FIG. 43Bshows as a comparison the T″ cross-section as averaged along the x-axis,obtained by including all f points along the x axis in the D window usedfor FIG. 43B. Such a 1D-extended window choice makes FIG. 43B (or, moreprecisely, the x-axis of FIG. 43B) a reasonable choice for T″ in thecase of 1D patterns (though it should be noted that FIG. 43B iscalculated with a different field size and gridding from the example 1DT″ plot previously exhibited in FIG. 32, and in addition FIG. 43B uses adifferent choice for parameter p). FIG. 43B thus employs a cross-sectionaveraging that is suitable for constructing T″ in a 1D pattern context,and as such the FIG. 43B T″ kernel might be applied in practice toparticular 1D frequencies like that used in the single-frequencycross-section of FIG. 43A (which is a worst-case 1D frequency in termsof TCC^((r)) error). The FIG. 43B kernel will be scaled to exactly matchthe peak point of the fin at the FIG. 43A frequency (since theloxicoherent structure guarantees this for all peak frequencies, inpreferred embodiments), but will only fit the off-peak TCC^((r)) in thetwo directions (Δf_(x) and Δf_(y)) orthogonal to this point to theextent that the FIG. 43B cross-section has a similar shape to thespecific local cross-section of FIG. 43A.

FIG. 43B is in fact seen to bear a general resemblance to thesingle-frequency cross-section of FIG. 43A, as expected from theGibbs-like behavior involved, but FIG. 43B does show non-negligibledifferences in its fine structure. However, in this context our earlierhighly successful 1D pattern results (e.g. FIG. 28) indicate veryfavorable prospects for 2D correction, since our earlier results haveshown that a T″ kernel calculated by using a wide 1D window in FIG. 22J(this window being broadly equivalent for purposes of discussion to thatused in FIG. 43B) is able to very substantially reduce the image errorat all 1D spatial frequencies, including the frequency of worst-caseerror at which the un-averaged FIG. 43A cross-section was extracted. Inother words, the performance results reported in FIG. 28 illustrate howwell an x-averaged T″ can improve image accuracy at every individualfrequency along the x axis. This robustness is primarily the result ofsuccessfully reproducing the strong central peak of the cross-section.However, besides reproducing this central peak, the averaged FIG. 43Bfine-structure away from the peak is also seen to retain anon-negligible degree of correlation with the peripheral fine-structurein TCC^((r)) cross-sections at specific problematic frequencies likethat of FIG. 43A. This partial matching in the periphery of the fincross-section improves the accuracy of the loxicoherent system beyondthat obtained by matching the central peak, even though the averaging inFIG. 43B has e.g. blurred the definition of the diagonal arms that aremore prominent in FIG. 43A than in FIG. 43B, and has removed some of theazimuthal variation seen in the FIG. 43A peripheral rings. Despite thisblurring of the periphery from averaging, the use of a wide enough Dwindow to give T″ coverage over the entire (1D) band limit is found togive good results at every specific frequency within the window.

FIG. 43C shows the T″ result obtained using a 2D averaging window whoseshape has been specialized for explanatory purposes. In particular, the2D window used to obtain FIG. 43C only extends across the full bandwidthof f in the radial direction, while azimuthally the window function Dmerely spans an arc of 22.5°. More specifically, the averaging window Din FIG. 43C is a 2D pie-slice whose upper radial boundary is the x-axis,and whose lower radial boundary extends into the third quadrant (i.e.towards negative y) through an angle of 22.5°. The mean orientation ofthis window has thus been rotated out of alignment with the x axis by12.25°, for reasons that will be discussed. The T″ function in FIG. 43Ccontinues to show the same key central peak as FIGS. 43A and 43B.Moreover, despite being more extensively averaged, the peripheralcontent in FIG. 43C is for the most part only moderately attenuatedrelative to that in FIG. 43B, and the shape of this peripheral contentremains fairly similar. We have seen that when T″ for the firstloxicoherent system is obtained by averaging over a 1D window (as inFIG. 43B), T″ nonetheless proves able to correct a very large portion ofthe TCC residual error at every 1D spatial frequency (including theworst-case frequency of the FIG. 43A un-averaged cross-section). Thisimplies that the variation in T″ across these 1D frequencies, asexemplified by the noticeable deviation of the periphery in FIG. 43Afrom the FIG. 43B averaged result, is not large enough to eliminate thestrong correction provided by an averaged T″ function. The FIG. 43A to43B variation is larger than the change between FIGS. 43B and 43C, i.e.larger than the change incurred in expanding the 1D linear widow used inFIG. 43B to fill the 2D pie-slice used in FIG. 43C. Broadly speaking,this indicates high potential accuracy during 2D correction with asingle T″ kernel. However, the window used in FIG. 43C only covers a22.5° azimuthal range, and close inspection of FIG. 43C shows that theweaker off-peak content is rotated slightly (clockwise) relative to thatin FIG. 43B, due to the rotated orientation of the D window away fromthe x axis in the former case. (Of course, the difference between FIGS.43C and 43B is not purely one of azimuthal rotation, but this rotationis the only observed consequence of the expanded averaging used in FIG.43C that has no analogue in 1D expanded averaging.) In FIG. 43C thisazimuthal change is only a weak cosinusoidal effect, but such anazimuthal averaging would be more significant if the D window wereextended to cover a full 180° half-plane, or a full 90° quadrant,although extraction of the dominant central peak would not be greatlyaffected by a wider averaging. (Lithographic sources are usuallybilaterally symmetric, and the c-quad source example used here is alsosymmetric between x and y.)

The observed azimuthal variation is a consequence of the wider set ofpositional combinations available in the 4D TCC, since it is notpossible for a single T″ function to cover these combinations ascomprehensively as is possible in the case of 1D patterns (2D TCC), eventhough T′ and T″ do increase their dimensionality to become 2D functionswhen applied to 2D patterns. For example, a slice of the 2D T″ that istaken along the x axis will function in the role of a one-dimensional T″kernel for the set of 1D patterns in which f is oriented along the xaxis [with the Δf axis of T″(Δf) then being parallel in terms of x,yazimuth to the 1D frequency axis f of these patterns, i.e. bothfrequency arguments are oriented along x since the 1D patterns arevertically extended], but this same one-dimensional T″ slice must alsoprovide an axis of correction for f displacements along othernon-parallel azimuths. For small changes in azimuth the associatedvariation is only cosinusoidal, i.e. quadratic, and is therefore small,but it becomes more significant over the full angular domain.

Based on these considerations, one exemplary approach for generalizingthe loxicoherent correction to 2D has four basic elements, namely: finsectoring, exploitation of intermediate-range spatial homogeneity in ICpatterns, weights to exploit frequency domain inhomogeneity, and use ofhigher-order loxicoherent systems to cover a broad azimuth in the 2Dmask plane. These four elements will now be discussed in more detail.

1) Fin Sectoring:

By azimuthally sectoring the fin shape function it is possible to almostentirely recover in 2D the strong 1D performance demonstrated above, bycalculating separate T″ kernels that correct one azimuthal sector of thefin at a time, although this approach may require as many as fourloxicoherent systems to achieve about the same jump in accuracy as eachsingle loxicoherent system in the 1D case. While sectoring may notalways be the most efficient 2D strategy, it has the advantage of beingable to provide a clear extrapolation from 1D to 2D in the form of the4-for-1 substitution that occurs when the full azimuth is divided into 4sectors, as will now be discussed.

Suitably sectored kernels may be obtained by calculating T″ usingsector-shaped window functions D in FIG. 22J, with these sectorspreferably being pie-slice-shaped regions in the {f_(x), f_(y)} domainin order to minimize the azimuthal variation across the sector breadth.While FIG. 43C was generated using an asymmetrically oriented sector inorder to highlight the azimuthal variation effect, actual mitigation ofthis azimuthal variation is preferably accomplished using sectors thatare symmetrically spread and oriented relative to the x and y axes.Since the azimuthal effect will tend to introduce only a slow,cosinusoidal variation across a sector, it is possible to employ sectorsof non-negligible angular breadth without incurring a severe azimuthalblurring of the fine structure. For example, FIG. 43C has shown thataveraging over a sector extending below the x axis by 22.5° will onlygive rise to a small skew in orientation into the third quadrant,indicating that a symmetrically oriented sector extending for ±22.5°about the x axis (i.e. with a total width of 45°) will only give rise toa mild azimuthal blurring. Since lithographic sources are oftenconsidered to have bilateral symmetry during the operations where OCS isemployed, each sector may be extended along both positive and negativeradial directions (and asymmetries are usually small even when included,making the averaging from a positive/negative-extended sector windowacceptable in the context of a loxicoherent contribution whose totalmagnitude is itself a small fraction of the overall intensity). As willbe discussed, each such sector effectively gives rise to its ownloxicoherent system under this procedure. Since the azimuthal variationacross each individual sector is small, the four systems that exhaustthe full 360° fin azimuth will together accomplish much the same resultas a single 1D system of the FIG. 20H form. The computational overheadassociated with even these four sectored loxicoherent systems is farsmaller than would be needed to achieve the same accuracy gain usingadditional OCS kernels (see e.g. FIG. 28).

When the fin is sectored, the T″ kernel that is obtained with eachsingle sector window D(f) will, by construction, only extract TCC^((r))within the 4D domain D(f₁)D(f₂). The set of such domains obtained fromall sectors will be disjoint in the 4D space if the individual Dfunctions are disjoint in the 2D f subspace of the fin, as will be thecase when the fin is divided into separate azimuthal sectors. Moreover,even though the various D windows are applied to the same overall finstructure, i.e. potentially to a single common T′ function per FIG. 22G,the actual fin content T′×D within a sector will be disjoint from thefin content within every other sector when the windows are disjoint,making T′×D effectively equivalent to a sector-specific T′, i.e.creating separated effective T′ functions whose windowing boundaries arebuilt-in. This means that both the T′ and T″ functions for each sectorare equivalent to the kernels of a sector-specific loxicoherent systemthat is completely independent from the systems for the other sectors,with the ensemble of such disjoint systems serving to exhaust the 2D finin much the same way as does the first 1D loxicoherent system, though inthe full 4D space there are distant peripheral regions outside theensemble window formed by the D(f₁)D(f₂) products.

FIG. 44 shows an example of four effective T′ functions that areconstructed by the sectoring method, using windows D that select fromthe FIG. 41A fin (which per FIG. 22G is technically the square of T′throughout the bandpass) a quartet of appropriate azimuthal sectors thatare built into the corresponding disjoint T′ functions. Since the C-quadsource used in this example has x-y symmetry (i.e. mirror symmetry aboutthe ±45° diagonals), the kernels for the FIGS. 44B and 44D sectors canbe obtained by simple geometrical rotation of the kernels for the FIGS.44A and 44C sectors. Use of the FIG. 44 kernels in loxicoherent systemsto extract the dominant fin entails eight additional convolutions,rather than only the two convolutions that are needed for the 1D case.Per the discussion above in connection with FIG. 43, the use of onlyfour sectors to subdivide the azimuth is generally sufficient tostrongly mitigate the generic azimuthal dependence of T″ in the 4D case.However, if desired, a finer azimuthal segmentation may be used toreduce the quasi-cosinusoidal azimuthal variation seen in the 2D fincross-section to an arbitrarily low level. One trade-off is that two newconvolutions are needed for each added sector, although four sectorswill typically prove sufficient. (Sources that strongly depart from theusual bilateral symmetries may require more sectors, if the TCC [morespecifically TCC^((r))] has correspondingly strong asymmetries.) Since xand y oriented spatial frequencies are usually the most critical, onealso has the option of dispensing with the sectors for the ±45°orientations. Further, as will be discussed, it may be sufficient toreproduce only the narrow central spike in such cross-sections, and inaddition weights can be incorporated.

A further improvement may be obtained by jointly optimizing the T″kernels of all sectors collectively, instead of optimizing each T″kernel separately to minimize the TCC residual left by the loxicoherentsystem of its individual sector. This may be accomplished by proceduresthat are discussed in more detail below, including the use of windowfunctions D that do not fall entirely to zero outside their particularsector.

2) Exploit IC Pattern Intermediate-Range Spatial Homogeneity:

The local fin 2D cross-section has been seen to have a narrow centralspike whenever the fin peak is large, as a consequence of FIGS. 5 and 7.This central spike is ubiquitous throughout the full x,y azimuth of the2D fin; for example, it is seen to be present in FIGS. 43A, 43B, and43C. The peripheral structure is more variable, but has much lowermagnitude. A significant improvement over OCS can thus be obtained fromone added loxicoherent system that uses a single averaged T″ function tosuccessfully capture this narrow spike, even if the fine structurevaries azimuthally. Moreover, accuracy is in practice further aided bythe spatial scales involved. After the first loxicoherent system isextracted, the remaining TCC error will typically attain its greatestmagnitude at frequencies which are slightly offset from the removedcentral peak. TCC^((r)) content at these azimuthally varying small-Δffrequencies will determine the long-range falloff of the t″ function inthe spatial domain, and different azimuthal weightings will give rise todifferent falloff behaviors. Per FIG. 20H, this long-range falloffgoverns an intensity contribution at image position x that is given by a(relatively) long-range integration over the neighborhood of x withkernel t″. Here “long-range” should be understood as being comparable tothe so-called optical diameter, which might typically be of order 2microns. In the general context of advanced IC design such a range mightbe regarded as only “intermediate” in scale, but it is still many timeslarger than critical IC feature spacings, and in modern IC layouts thefeature content in a given layout is often required to be fairlyhomogenous (with much repetitive content, and with many varieties ofshape constructs being prohibited in order to ensure printability).Moreover, IC layout content is nowadays usually required to be uniformin density at intermediate scales, with “dummy” fill features beingadded where necessary to improve uniformity. This pattern homogeneityand density uniformity makes the loxicoherent contributions in FIG. 20Hless sensitive to modest changes in the shape and scale of theintegrating t″ kernel, so long as t″ remains large compared to the sizeand spacing of typical IC features. Layout homogeneity and uniformity atintermediate scale therefore reduce the impact of variations in theshape of the central fin spike, and thus improve the practical accuracyprovided by a single averaged T″ rendition.

3) Weights to Exploit Frequency-Domain Inhomogeneity:

The above-mentioned intermediate-scale spatial homogeneity in modern IClayouts often manifests itself as repetitions and near-repetitions ofpreferred device constructs, and in many cases these repeated patternsare at least mildly extended in either the horizontal or verticaldirection in the layout, even though the patterns involved usuallycannot be regarded as fully one-dimensional. Both tendencies give riseto a strong concentration of energy into spatial frequencies that liealong the f_(x) and f_(y) axes. Moreover, there will usually be certainfrequency harmonics along x and y that are strongly favored in a givenlayout, while others will only contain weaker residual content, due e.g.to “forbidden pitch” design rules. Both kinds of inhomogeneity in thediffracted spectrum are clearly seen in the FIG. 27B example, and theparticular metal level clips that give rise to the FIG. 27B examplespectrum turn out to exhibit more diversity than is found in most IClevels. In many cases the electrically consequential edges andseparations that define the so-called critical dimensions of the IC arelikewise aligned with the x and y axes. With some layouts the ±45°meridians are also prominent in the diffracted spectrum.

When T″ kernels of broad (or full 360°) azimuthal coverage are beingemployed, this strong variability in pattern significance can beexploited by using D windows that are not binary. In other words, ratherthan merely sectoring the TCC^((r)) domain using binary D functions withvalue 1 inside the sector and 0 outside, the function D(f) can givepreferential weight to spatial frequencies along preferred design axes,thereby achieving greater accuracy in the calculated intensitycomponents that are of greatest practical interest. In many cases it ispreferable to apply weights of this kind in combination with the item 1sectoring method listed above. The advantage provided by the weights issynergistically enhanced in situations where the item 2 benefit fromintermediate-range spatial homogeneity is also present. D functions canalso be modified to exhibit stronger correlation with circuit designconstructs that are deemed particularly important (e.g. critical pitchesas a simple example). Note that window function D was introduced toprevent ill-conditioning in the T″ solution, whereas the weights beingdescribed here are intended to ensure that azimuthal blur is minimal atspatial frequencies that make a critical contribution to pattern images.Referring to FIG. 45, we may distinguish these different purposes byintroducing a separate weighting function Γ(f₁,f₂) that is distinct fromwindow D, as shown in FIG. 45. One suitable form for Γ(f₁,f₂) is thegeometric mean of the expected pattern energy levels at frequencies f₁and f₂ (obtained, for example, from sample clips, as in the FIG. 27example), added to a uniform base weight.

D windows of more complicated form than simple 0-or-1 indicatorfunctions (i.e. more complicated than simple apertures which fullyseparate the different sectors) are also beneficial when jointlyoptimizing the T″ kernels of different sectors together, as will bediscussed.

4) Address Broad Azimuth Using Higher-Order Loxicoherent Systems:

The azimuthal T″ sectors described above in item 1 of the present listinvolve the use of multiple loxicoherent systems to thoroughly exhaustthe dimensionally expanded domain of averaging that must be accommodatedduring calculation of a T″ kernel for 2D patterns, avoiding the need tocover this content with a single D window that extends over the full360° azimuth (with associated azimuthal rotation/blurring of peripheralcross-section content). While the sectoring method is explicitlydesigned to address the azimuthal averaging effect, other more generalmethods can be employed to extract TCC^((r)) content over the full 360°azimuth using multiple loxicoherent systems. In particular, a series ofhigher order loxicoherent systems can be obtained by successiveapplications of the homotopy method discussed in connection with FIG.37, or by the methods discussed in connection with FIGS. 34 and 35. Allof these methods apply in 2D as well as 1D.

The four methods just described for handling the added dimension ofaveraging that arises in calculating the T″ kernel for 2D patterns canall be used in conjunction with one another. This illustrates a moregeneral point, namely that the various forms of loxicoherent systemdisclosed in the invention [e.g., fin-targeted to extract the Δf=0dominant residual (FIG. 22), higher-order systems to target non-DCdiagonals (FIG. 34 or 35), homotopy-derived (FIG. 37), DC-monolinear(FIG. 41), azimuthal fin-sectored (FIG. 44), or content-based frequencyweighted (FIG. 45)] can fruitfully be used in combination with oneanother, as well as in conjunction with N standard OCS systems asdescribed in FIGS. 20H and 20I. In these various loxicoherent systemsthe T″ constituent incoherent kernels are usually chosen to provide aleast-squares optimal reduction in TCC^((r)) over the doubled domain, ashas been discussed. In a preferred embodiment the P kernels ofDC-monolinear systems are also chosen in this way.

An additional refinement when multiple loxicoherent systems are employedis to improve accuracy by choosing the multiple T″ and P kernels to bejointly optimal in a least-squares sense, rather than merely optimizingthem for the performance of the individual systems in which they areconstituent. Such a procedure may be further understood with referenceto FIG. 46, which considers as an example the joint least-squaresoptimization of the T″ kernels in two loxicoherent systems that togetherattempt to match TCC^((r)), with these two systems being designated “A”and “B”. In this simple example the two systems A and B both employ thebasic structure of FIG. 20B, but it will be clear to those skilled inthe art how the procedure can be extended to cover more complex systemembodiments, and the use of more than two loxicoherent systems.

For the simple two-system example, the equation of FIG. 46A in FIG. 46expresses the total squared error (denoted E_(Multisystem)) in matchingTCC^((r)). The squared error is summed over the full doubled domain, butthe error at different frequency pairs may be given different weightsduring this integration; weights are defined by the weighting function Γshown in the first line of FIG. 46A, as has been discussed previously inconnection with FIG. 45. The second line of FIG. 46A shows the residualTCC that is being matched by the A and B pair of loxicoherent systems,and these two systems themselves are shown in the third line (inbrackets). The T′ constituent coherent kernels for the A and B systemscan include window functions (denoted D in previous equations), butthese are not explicitly shown in FIG. 46A.

The constituent incoherent systems T_(A) and T_(B) can be chosen tominimize E_(Multisystem) by steps very similar to those used to obtainFIG. 22J. In order to display the resulting solutions for T_(A)″ andT_(B)″, it is convenient to introduce two simplifying notations. First,FIG. 46B introduces the symbol {circumflex over (T)}(f,Δf) to representcomputationally a constituent coherent system T′(f₁)T′*(f₂). In thisexample two such systems are present, namely {circumflex over (T)}_(A)and {circumflex over (T)}_(B). (Note that coherent systems {circumflexover (T)} are not equivalent to rotated system kernels

, though they are represented by visually similar symbols.) Second, FIG.46C introduces a kind of weighted dot product notation involving theintegration of multiplied functions along the f axis in the doubleddomain. In particular, for two arbitrary frequency-domain functionsdenoted F and G, whose arguments span the full dimensionality of thedoubled domain (e.g. f₁ and f₂, or f and Δf), this dot product (usingweight function Γ) is denoted F∘G, and is defined by the integral on theright side of FIG. 46C. Using this notation, the optimal values forT_(A)″ and T_(B)″ at each difference frequency are given by the solutionto the pair of ordinary linear simultaneous equations shown in FIG. 46A.These equations generalize in the obvious way to increased numbers ofjointly optimized systems, and can of course be solved very rapidly toobtain values for T_(A)″ and T_(B)″ on a sequence of grid points.

Joint optimization of T″ kernels can be useful when azimuthal sectoringis employed (e.g., as discussed in connection with FIG. 44). Forexample, it can be advantageous to jointly optimize the four T″ kernelsin the loxicoherent systems of the sectors shown in FIGS. 44A-44D byusing the four-variable analogue of FIG. 46A, but in doing so one shouldmake a modification to the windows that define these sectors. Inparticular, the sector windows should be modified in a way that preventssingularities in the simultaneous equations of FIG. 46A (extended to thefour variable case), for example by using D window functions for eachsector that do not fall entirely to zero outside the sector boundaries.As a more specific non-limiting example, instead of entirely suppressingto zero the TCC^((r)) content outside the sector, as do the simpleaperture-like windows depicted in FIGS. 44A-44D, the D functions chosenfor joint sector optimization might provide a non-disjoint aperturing ofTCC^((r)) to a relative value of e.g. ⅕ outside their sector, whilecontinuing to have full value within their sector. D should then berescaled in order that the 2D fin peak be properly divided by the foursectors in combination, and since each sector contains a product of twoD functions, an interior-to-exterior ratio of 5:1 requires that the Dwindow have transmission √{square root over (⅝)} within the sector, and√{square root over (⅛)} outside the sector. When using this embodimentit is also useful to include a weighting function Γ, of the kinddiscussed in connection with FIG. 45.

The various novel decomposition systems described herein may all beoptionally improved with numerical refinement. This would typically usethe T′ as optimization variables (e.g. in a gridded representation),with optimization being applied either to individual systems insequence, or to multiple systems jointly. During each iteration of suchan optimization the T″ kernels may be set using e.g. FIG. 45 or 46D.

Beyond the FIG. 46 method or other numerical refinement steps tooptimally employ the multiple kernels and systems of the inventionjointly, a synergistic combination of the loxicoherent systems withadditional variant coherent systems may also be desirable. The potentialgain here need not be hampered by the fact that OCS exhaustion of theTCC function inevitably becomes slower after the initial OCS kernelshave been extracted. This is because the additional variant coherentkernels can be generated, in accordance with the invention, from theTCC^((r)) that remains after at least one loxicoherent system hasalready been extracted, resulting in a TCC^((r)) for extraction of newcoherent kernels which lacks the dominant fin that is present whenadditional coherent kernels are found to provide only a diminishedbenefit during prior art OCS. Per the discussion of FIG. 29 above, thisfin has been shown to present an obstacle to the efficient extraction ofcoherent systems in the context of the standard OCS procedure.Subtraction of the fin from the residual may constitute a “disruptivereset” of TCC^((r)) to a condition that no longer approaches theOCS-unproductive asymptotic regime described in FIG. 29. Thus, in somecases it may be possible to achieve an additional rapid increase inconvergence by re-diagonalizing the residual after theloxicoherent-targeted structure has been removed, and then employing theresulting coherent kernels as new OCS terms. However, it should be notedthat the TCC residual will generally not be positive semi-definite afterthe fin is removed. In practice an optimum strategy can involve a mix ofmultiple kernel types, both new and traditional.

Another variant of the loxicoherent methodology that can be implementedin the invention is the use of heuristic kernels in the loxicoherentsystems employed, or the use of kernels that are empirically calibrated.Such kernels may be used either as an alternative to the kernels derivedfrom physical optics that have been disclosed thus far, or as asupplement to the physical kernels.

As discussed above, it is standard practice during OPC to account forresist effects using model forms whose coefficients and parameters areempirically calibrated. The individual terms of these models are oftenchosen on quasi-heuristic or phenomenological grounds to mimic the knowncharacteristics of physical processes that take place within thephotoresist, e.g. using convolutions of Gaussian kernels with theexposing image to mimic the diffusion of acid catalysts within theexposed resist. The loxicoherent systems described thus far have beenchosen to match the physical TCC, e.g., to match the parts of thephysical TCC that conventional OCS systems do not capture. However,phenomenological parametric adjustments can be added to these kernelfunctions, with the parameters and weights being set during resist modelcalibration. Loxicoherent systems using adjustable kernels of a purelyphenomenological character can also be deployed by means of such acalibration procedure, and the calibration may be made againstrigorously calculated optical images instead of resist measurements.

In the embodiments discussed thus far, the invention employs a highlyefficient means to calculate the intensity of lithographic images. Indoing so it maintains a computational scaling that is almost linear withthe area of the image field, this scaling being dictated by thenear-linear scaling of FFTs. Mask design tools that use the prior artOCS method also achieve near-linear scaling with area, and are likewiselimited by the scaling of FFTs, but use of loxicoherent systems makesthe overall constant of proportionality for this scaling considerablysmaller, due to the greatly reduced number of convolutions that areneeded to achieve a given accuracy level in the calculated image.

Thus far the invention has generally been discussed in the context ofembodiments that provide mask shapes whose dimensional compensation isdetermined by OPC. Ordinarily, the only optical calculation withsignificant computational cost that must be undertaken during OPC is thedetermination of image intensity. This is because OPC typically carriesout mask adjustments using a low-cost iterated feedback scheme. As hasbeen discussed, such a feedback scheme typically involves either closingadjustable mask edge fragments inward, or extending them outward,according to whether the intensity at sampled positions along the targetcontour sought for the printed image either exceeds, or falls short of,the anchor value, with the sampled positions being e.g. the points alongthe target contour that are closest to the midpoints of the adjustablefragments.

However, in current practice an alternative to OPC has gained popularityas a way to design lithographic masks that can print IC shapes withgreater fidelity and with decreased sensitivity to processingfluctuations, namely the class of methods sometimes referred to asInverse Lithography Technology (ILT), or as mask optimization. ILTobtains lithographic mask designs as the solution to a formal nonlinearoptimization problem (or sequence of such problems), in which theoptimization variables that define the mask edge positions are notsimply tied to individual neighboring intensity sample points, as inOPC, but are instead numerically optimized in terms of their collectiveglobal impact on lithographic constraints and objectives, i.e. mask edgevariables are optimized against quantitative metrics that expresslithographic goals or requirements. These constraints and objectives mayreflect standard lithographic metrics, or they may have a morephenomenological character, e.g., being barrier terms that downweightundesirable image conditions. In many cases these metrics are derivedfrom the exposing intensity, though some metrics may express maskmanufacturability limitations by directly downweighting or penalizingunresolvable separations (spacings) between edge variables at the masklevel, or by constraining these unmanufacturable spacings out.

When choosing the adjustments that will be made to an interim masksolution in the next iteration, feedback-based methods (like those usedin OPC) are guided by the intensity pattern that the interim solutionproduces (or they may be guided by the resist response that is itselfdriven by the intensity pattern). However, optimization-based methodstypically make use of gradient information as well; in other words, themask adjustments made during ILT are typically driven by the derivativesof intensity-based constraints and metrics with respect to the problemvariables, as well as by the constraint and metric values themselves. Asfar as the latter non-derivative inputs are concerned, it will beassumed that the bottleneck computational cost involved in calculatingthe intensity-driven metric and constraint values is the basic cost ofcalculating the intensity image itself, though in practice theintensity-based resist models that may be used during both OPC and ILTentail non-negligible cost. Nonetheless, these resist-model computecosts are common to both prior art OCS decomposition and the novelloxicoherent decompositions employed by the present invention, and areusually smaller than the cost of the common intensity calculation.Though some resist model regression functions are, like the opticalmodel, gated in compute time by frame-scale convolutions, theconvolution count in such resist models is fairly small, e.g. 5 to 10.The basic step of calculating the optical image is often thecomputational bottleneck, both in OPC, and in evaluation of thelithographic metrics that are the basis of ILT optimizationformulations. The present invention can significantly improve the speedand accuracy of this calculation, for either OPC or ILT, achieving anear-linear scaling with the area of the simulation frame.

However, one consideration for carrying out ILT in the presentinvention, which does not arise with OPC, is that methods for solvingnonlinear optimization problems typically employ derivative informationto adjust the variables, i.e. gradients. Use of derivatives would seemat first glance to entail the computation of a considerably largervolume of information than mere computation of the intensity-drivenlithography metrics alone, since these derivatives must be taken withrespect to every mask variable, and the number of mask variables isitself proportional to problem area (with the cost of computing eachsuch derivative being of the same order as the compute cost of themetrics themselves). Fortunately, as will now be discussed, there areembodiments of the invention that can obtain the derivatives withrespect to all problem variables (of the relevant loxicoherentcontributions to quantities of interest) with near-linear area scalingoverall. Using methods already known in the art, the invention can alsoobtain with near-linear scaling the necessary derivative information forthe contributions made by the coherent system set. The loxicoherentsystems used by the invention can significantly reduce the total numberof decomposition systems that are needed to meet a given accuracytarget, thereby significantly reducing the overall compute time requiredfor ILT mask design.

This desirable scaling is achievable with loxicoherent systems under abroad range of approaches to optimized mask design. One such approachformulates the optimization problem as the unconstrained minimization ofa so-called “cost function”, where this cost function is typically aheuristic amalgamation of a number of diverse lithographic desideratum,these latter being expressed as either metrics of solution quality,and/or as penalties on conditions of lithographic print failure orexcessive process sensitivity. In another inverse lithography approachthe optimization problem is formulated as one of constrainedmaximization (or minimization), in which one lithographic quality metricis designated as an objective to be maximized subject to constraintsinvolving other lithographic metrics, with each such constraintreflecting a different lithographic requirement (or the constraint mayreflect a single specific application of a general requirement at one ofmultiple locations in the image, each such location generating its ownconstraint to express the requirement). The overall mask design processcan also involve solving a sequence of optimization problems in whichthe constraint set and objective choice are changed from problem toproblem.

Mask optimization generally requires a larger number of adjustmentiterations to converge than does OPC, particularly when the moresophisticated ILT formulations and flows are used. In OPC procedures thelengths of adjustable edge fragments are usually held fixed, with thelengths of the interleaved connector edges then being automaticallydetermined once the excursions/retractions of the adjustable edgefragments have been set. With this OPC methodology the connector edgesare not shifted in and out, and thus the connector edges do notrepresent adjustable degrees of freedom during OPC. In contrast, ILTformulations typically allow the lengths and positions of all mask edgesto change during mask optimization, resulting in a higher variable countthan in OPC. The gain in lithographic image quality that comes fromdesigning mask shapes with ILT instead of OPC is often increased whenvery fine fragments are used, i.e. when edges are more denselyfragmented than is useful with OPC; however, fine fragmentation furtherincreases variable density. For these reasons ILT formulations arenumerically more intensive than OPC, with constrained ILT formulationsbeing particularly costly in compute time. Because of these highercompute costs, ILT is not commonly applied at full chip scale, andapplications of ILT over relatively large areas will typically usesimpler cost function formulations.

The above-described aspects of the cost function and constrainedformulation approaches are known in the prior art. In general, at leastsome of the lithographic metrics involved in either approach arenonlinear in the problem variables, placing these optimization tasks inthe category of so-called nonlinear programming problems.

Constrained nonlinear optimization problems are formally morecomplicated than unconstrained problems. However, many methods forconstrained nonlinear optimization include inner loops in which aninterim combined quantity is maximized in an unconstrained way withrespect to the problem variables, with this interim combined quantitybeing formed as an amalgamation of the problem objective with barrier orpenalty terms that initially serve as approximate surrogates for theproblem constraints, with these barrier or penalty terms then beingevolved during optimization (in outer loops) to ultimately enforce theconstraint requirements in an exact way. During the inner loops thecombined quantity is maximized without explicit constraint (with theexception that some methods explicitly impose the variable bounds duringthis maximization).

The inner loop maximizations are carried out on the basis of calculatedgradients, with second-order information often being built-up andexploited during the course of maximization by means of e.g. theso-called BFGS method. (This inner-loop maximization is operationallyalmost identical to minimization of a cost function.) Then, during outerloops, the penalty or barrier terms are adjusted on the basis of thepreceding maximization results in such a way as to ensure thatsucceeding unconstrained maximization solutions conform steadily moreclosely to the solution of the true constrained optimization problem.Convergence may be achieved, for example, if the combined quantityevolves to equivalence with a maximized Lagrangian of the standard kind,i.e. if the penalties vanish for constraints that are not binding, butact as constraint terms in an ordinary Lagrangian for those constraintswhich are binding on the solution.

For example, when the so-called Augmented Lagrangian method is used tosolve a constrained optimization problem, the combined quantity to bemaximized may be formed by adding certain quadratic terms to the usuallinear constraint terms of an ordinary Lagrangian, in order to improveconvergence. Referring to FIG. 47, the Augmented Lagrangian (AL) may ingeneral be defined according to FIG. 47A, where C₀ denotes the objectiveto be maximized, and where P is shaped in a specialized way that alsoexpresses a penalty or barrier. (It will be clear to those skilled inthe art that this meaning for the symbol “P” as used in FIG. 47 is verydifferent from that discussed in connection with FIG. 41, where “P”designated a window function or kernel in a DC-monolinear system.) Theith penalty term P_(i) in the AL sum is associated with the ithconstraint in the problem formulation. Qualitatively, P_(i) detractsfrom the AL when the ith constraint c_(i)({right arrow over (e)}) isviolated, or more precisely (particularly when the outer loops have notyet approached convergence), when the ith constraint is merely close tobeing violated. Here {right arrow over (e)} denotes the list of problemvariables, which may, for example, comprise a list of horizontal andvertical coordinates of adjustable edge fragments in polygonal maskopenings. For purposes of explanation it may be supposed that the maskpolygons are of the so-called Manhattan kind, in which all edges areeither vertical or horizontal. In the context of ILT optimization onecan, for simplicity, use the term “edges” to refer to both theadjustable edge fragments in the mask polygons, and to the feature edgesof the IC design shapes. As discussed, adjustment of the edge positionvariables will cause changes in the mask edge spacings and separations,i.e., the dimensions and separations of the mask features (and thereforetheir relative positions) will be adjusted, allowing these shapes to bedimensionally compensated. Assume that λ_(i) is the current estimate ofthe Lagrange multiplier for the ith constraint, and v_(i) is a penaltyparameter. The λ_(i) and v_(i) are updated during outer loop iterations.Specific procedures for updating and initializing the multiplierestimates and penalty parameters are discussed in D. P. Bertsekas,Nonlinear Programming (Athena Scientific, 1995), chapter 4. Bertsekasshows in detail how the AL may be structured in such a way that it canbe evolved to a true Lagrangian when outer loop convergence is complete,at which point the AL will become equivalent to a standard Lagrangianwith valid λ_(i) multipliers. In practice the solution may be deemedacceptable before full convergence has occurred.

A convenient convention in such a procedure is to reformulate allconstraints c_(i) to have 0 as the acceptable lower limit, as shown inFIG. 47B (e.g. by inclusion of a constant offset in the definition ofeach c_(i), if needed). When the constraint functions c_(i) areformulated in this way, a suitable structure for the penalty function Pis given in FIG. 47C. Once c_(i) is known, P_(i) can be evaluated viaFIG. 47C at almost no computational cost; thus the main computationalcost in evaluating the ith term in the FIG. 47A sum lies in evaluatingc_(i). In general, the key subset of the problem constraints thatexpress overall lithographic performance requirements in terms ofoptical intensity can generally be cast as a list of constraints in theFIG. 47B form, whose number will scale linearly with the total mask areabeing optimized. Moreover, each single such constraint can generally bederived from the optical intensity at a small fixed number of points,and with only a small fixed number of operations being needed toevaluate c_(i) once these intensities are known. Those of ordinary skillin the art will appreciate that many standard lithographic requirementsand objectives can be formulated in this way, such as constraints onimage slope, integrated process window, MEEF sensitivity, andsuppression of bright or dark printing artifacts. These constraints maybe supplemented by a small quasi-fixed number of constraints that eachdepend on a larger number of intensities, but in general the totalconstraint count will be proportional to the mask area being optimizedin the current optimization run (which would typically be the area of asimulation frame). Additional geometrical constraints not involving theintensity may be included to ensure that the final mask solution ismanufacturable, and these constraints (along with the contributions oftheir derivatives to the AL gradient) may be obtained with near-linearscaling using, for example, the method described in U.S. Pat. No.8,719,735, Optimizing Lithographic Mask for Manufacturability inEfficient Manner, M. Sakamoto et al., incorporated by reference herein.

When the optimization problem is formulated using constraints of thiskind, the time needed to evaluate the sum in the FIG. 47A AugmentedLagrangian will scale linearly with mask area, once the intensity atrelevant sample points is known. Objectives to serve as C₀ can beformulated to express a wide variety of quality metrics, and in mostcases C₀ can be evaluated in a compute time that, at worst, scalesnear-linearly with mask area. Examples of objectives with this desirablescaling include integrated process window, RMS exposure latitude, andworst-case MEEF. In general, worst-case or first-to-fail metrics thatinvolve gating patterns are common in lithography, and these canformulated via the c_(i) constraints and the C₀ objective by usinginfinity-norms, which can be introduced into constrained formulationsusing well-known methods involving auxiliary variables.

Further reference in this regard can be made to U.S. patent applicationSer. No. 14/185,440, filed Feb. 20, 2014, “Mask That Provides ImprovedFocus Control Using Orthogonal Edges”, Jaione Tirapu Azpiroz, Alan E.Rosenbluth, Timothy A. Brunner, (now U.S. Pat. No. 9,310,674, issuedApr. 12, 2016), incorporated by reference herein.

If the coherent and loxicoherent system counts (e.g. N and L in FIG. 22Hor 22I) are held fixed, the compute time needed to obtain the imageintensity across a regular grid of sample points (i.e. at the gridpointsof an intensity “bitmap”) will scale near-linearly with mask area, i.e.with the so-called N log N scaling of the FFTs used to carry out theFIG. 20H or 20I convolutions. (Here N is proportional to mask area; itis customary to use the symbol “N” in describing this FFT-based scaling,and it will be clear to those skilled in the art that this meaning of“N” is distinct from that used in e.g. FIG. 20H.) The intensity at anyoff-grid sample point can then be obtained to high accuracy byinterpolating to the bitmap using a short-range interpolating function,which entails only a small fixed number of operations. Thisinterpolating function can be a piecewise bilinear 2D triangle functionif linear interpolation is used, or a 2D form of the well-known Keys'cubic interpolator if cubic interpolation is chosen. In a typicalformulation the ith constraint may involve the intensity at a smallfixed number of sample points, and for purposes of explanation we willassume that the functional expression which combines these intensityvalues within the function c_(i)({right arrow over (e)}) is linear,though the method of the invention supports more general forms, as willbe discussed. Under these conditions c_(i) can be evaluated as aweighted sum of the intensity values of a small number of pixel valuesin the intensity bitmap, with these weights subsuming the necessaryinterpolation coefficients as well as the constraint's linearcoefficients.

The AL can then be written in the form shown in FIG. 47D, where the i=0term should be understood to represent the objective function, as willbe discussed. An index s has been introduced in FIG. 47D to identify theparticular bitmap pixel (i.e. gridded intensity value) to which a givenweight applies, with the weight taken on in the ith constraint by thesth intensity pixel being denoted f_(s,i). The range on the sum over sis listed as “sparse”, to indicate that few intensity pixels willparticipate in any single constraint, so that the sum can be evaluatedfor each constraint by considering only the small fixed number ofintensity pixels that are relevant to that particular constraint, withall of these pixels typically being in some small local neighborhood ofthe intensity bitmap. Evaluation of all the c_(i), and thus evaluationof the AL itself, can then be accomplished with linear area scaling,since the number of constraints scales linearly with area. Near-linearscaling of the overall AL evaluation is thus maintained when theintensity calculation step is included in the assessment, since it hasbeen established that the invention can obtain the intensity bitmapusing efficient loxicoherent systems with a compute cost exhibitingnear-linear scaling. It should be mentioned that FIG. 47D incorporatessome notational simplification for clarity that does not impact thisoverall conclusion. For example, the i=0 term in FIG. 47D should beunderstood to be the C₀ objective, with P₀ then being the identityfunction. FIG. 47D omits auxiliary variables for simplicity, as might beused to define infinity-norm metrics, and it considers onlyintensity-based constraints; however near-linear scaling can still beachieved when e.g. mask manufacturability constraints are included. Morespecifically, the AL may also include constraints to ensure that themask is manufacturable, e.g. that no parallel edges in the mask designare positioned so close to one another that the mask cannot befabricated. Such constraints and their calculation are described in U.S.Pat. No. 8,719,735, and the totality of these constraints (as well asthe associated gradient) may be calculated with near-linear scalingusing the methods described therein. While FIG. 47D is written with anominally linear dependence of the c_(i) on the sampled intensities,near-linear scaling is still achievable when the constraint functionsinclude a variety of nonlinear structural forms, e.g., if theconstituent intensities are passed through an analytic nonlineartransforming function to which the chain rule can be applied. Thisallows phenomenological resist models to be considered, for example.

Though the outer loops of the constrained nonlinear optimizationprocedure use, e.g., the Bertsekas method to adjust the λ_(i) and v_(i)multiplier and penalty parameters (in order to ensure that the objectiveis properly constrained at termination only by those c_(i) that aretruly binding), each interim optimization that is carried out during onecycle of inner loops will maximize the AL (as an unconstrained quantity)by adjusting only e.g. the mask and auxiliary variables, holding theλ_(i) and v_(i) fixed at their current values. Multiple adjustment stepsare taken in converging to the AL maximum during each cycle of innerloops, with each step (i.e. a set of changes to the variable values{right arrow over (e)}) being chosen based on a calculation of the ALand its gradient, and with second-derivative information also beingincorporated using e.g. a so-called BFGS estimate. Calculation of the ALgradient might appear to entail a more challenging scaling thancalculation of the AL itself, since calculation of the gradient requiresthe derivative of the AL with respect to each problem variable. However,it is known in the art that the AL gradient can be calculated withnear-linear area scaling if the intensity is calculated by OCS, and itwill now be shown how the contribution to the AL gradient from aloxicoherent system can likewise be obtained with a computational costthat scales near-linearly with mask area.

For simplicity the case of so-called Manhattan masks will be considered,in which the edges of the mask patterns have either a horizontal orvertical orientation against the x and y design axes of the IC layout.If the qth edge amongst all polygons is vertical, the associated maskvariable e_(q) will designate the x coordinate of the edge, and e_(q)will similarly designate the y coordinate if the qth edge is horizontal.For simplicity the discussion to follow will consider the case in whichthe qth edge is vertical. (As mentioned, the edges referred to in thisILT context are the edges of the polygons that are to be written on themask; these edges might be referred to as mask fragments or fragmentconnectors in an OPC context.) FIG. 47E expresses the contribution madeby a loxicoherent system to the derivative of the AL with respect tomask variable e_(q); in particular, FIG. 47E shows the contribution tothe qth element of the AL gradient that is made when a loxicoherentsystem contribution from FIG. 20H or 20I is used in the determination ofthe sampled intensities. Such an intensity sample may be denoted I_(s),i.e. I_(s) denotes the intensity I(x_(s)) at sample position x_(s).(Note that FIG. 47E follows our usual simplifying convention of omittingthe y coordinate for brevity when indicating position variables.) Inpractice the I_(s) sample values would preferably be obtained bydiscrete FFT-based calculations as discussed above, but it is useful tofirst present the loxicoherent convolutions as idealized continuousintegrals, and this has been done in FIG. 47E. The AL itself wouldpreferably be evaluated before its gradient, and subsequent to thisevaluation the derivative of all P_(i)'s with respect to their c_(i)argument (at their current c_(i) values) can be evaluated in a timeproportional to mask field area (once the intensity samples have beenobtained, at computational cost governed by FFT scaling), since theP_(i)'s are elementary quadratic functions. The ith such derivative ofthe penalty function itself (with respect to c_(i)) is denoted {dot over(P)}_(i).

The working mask solution (whose transmission is m(x)) couples into theFIG. 47E derivative via a convolution with the t′ kernel. As aderivative, FIG. 47E essentially gives (as a ratio) the differentialresponse of the AL (or more precisely the differential contribution ofthe loxicoherent system to the AL) that results from a differentialadjustment of the qth mask edge. If the mask polygons correspond toclear openings in an opaque film, a differential outward translation ofthe qth edge will introduce a new filament of transmission along theedge, and the resulting contribution to the convolution of t′ with themask content will consist of an added “sliver” or spike of shape t′ thatis centered at edge location e_(q). More specifically, when the 2Dcharacter of the polygon is taken into account, and supposing for thesake of clarity that the qth edge is vertical, the differentialcontribution to the convolution of t′(x,y) with m(x,y) as evaluated atoutput point {x_(k), γ_(k)} will be given by the integral with respectto y′ of e(x_(k)−e_(q), γ_(k)−y′), with y′ running along the length ofthe infinitesimally displaced edge. As previously noted, it isconvenient to suppress the explicit 2D character of the loxicoherentkernels, so t′ along the qth edge will be written as t′(x_(k)−e_(q)) forsimplicity. Since the loxicoherent system involves the absolute squareof the t′ convolution, its derivative will be proportional to twice thereal part of the product of this convolution with the differential edgeintegral, as shown in FIG. 47F. Current mask technologies provide anumber of different polarity options for the patterned mask films, i.e.for the transmissions of both the patterned apertures in the mask, andfor the mask background, though masks whose {aperture,background}transmissions closely approximate simple {1,0} binary levels are themost common type. To allow for arbitrary transmission options, FIG. 47Fincludes a factor Δτ that represents the change in point-transmission atan edge location when a mask edge is differentially translated outward.Further, a factor ξ_(q) is included to account for the fact that adifferential positive increase in the e_(q) variable may correspond toeither an inward retraction of the edge or to an outward excursion;ξ_(q) is thus +1 for rightward and upper edges, and −1 for leftward andlower edges. The AL itself should preferably be calculated before itsgradient, and this means that the convolution of t′ with m which appearsin FIG. 47F (evaluated at the point x=x_(k)) will already have beenevaluated when FIG. 47F is considered. As shown in FIG. 47G, it isconvenient to denote this convolution as K, folding in the factor Δτ*for simplicity.

The steps in FIGS. 47E-47G have been presented in terms of continuousconvolutions and associated integrals, but at this point it isconvenient to discretize these integrals as quadrature summations acrossthe same grid that is used to calculate the AL. FIG. 47H does this, forexample replacing the FIG. 47G continuous integral (having integrationvariable y′) of kernel t′(x,y′) [which is written as t′(x) forsimplicity] along the qth edge by a summation of gridded t′ values thatare weighted by quadrature coefficients g, using a summation indexdenoted r. Since t′ is smooth, its value at any point along the qth edgecan be accurately interpolated as a weighted sum of the t′ values atnearby gridpoints, with closely neighboring points within the edgemaking use of the same set of gridpoints for this interpolation. Inparticular, the same set of t′ gridpoint values will be used in theshort-range interpolation throughout the entire subset of points alongthe edge which share the same nearest pixel boundary (considering allpixel boundaries that the edge crosses). All points within each suchsubset share the same set of nearest gridpoint neighbors, so that anydifferences that exist between the interpolated values of t′ at thesedifferent points in the subset will be due entirely to changes in theinterpolation weights from point to point. In FIG. 47H the edge integralis approximated as a summation over r, and this sum can be formed bygrouping sets of points that share the same nearest pixel boundary, andthen integrating the varying interpolation coefficients along the edgesegment for each group. Alternatively, the quadrature coefficients g cancombine conventional quadrature weights with the interpolation weightsfor all gridpoints near the edge which contribute to theseinterpolations. The gridpoints involved are specified by index r.

Without loss of generality, it can be assumed that the summation over ris sparse and independent of mask field area, which strictly speakingwould mean assuming that all edges are bounded in length by some fixednumber of grid pixels that is independent of the size of the simulationfield. Edge lengths (which in an OPC context would be referred to asedge fragment lengths) are in fact typically made finer than theresolution of the exposure tool, in order to ensure that the position ofthe developed resist edge is finely controllable along its contour viaadjustment of the mask edge segments (i.e. fragments). This means thatthe range of r can be assumed to have a relatively short upper limitsince the range of r applies only to an individual mask segment, or moregenerally that the length of edge q (and therefore the range of r) willnot (on average) increase with the total area of the mask region beingoptimized. An exception to this general rule may arise with patternregions that are entirely one-dimensional over an extended distance, inwhich case mask edges may preferably be highly extended, so that theirlength may even exceed the width of the optimization frame. However, insuch situations the total number of edges will increase sub-linearlywith area, leaving the total computational burden unchanged from that inthe more usual scenario. For simplicity FIG. 47H assumes a short edgelength, and so writes the limit on the r sum as “sparse”, indicatingsparse coverage and a limited total count.

FIG. 47H is nominally eight dimensional, in that it consists of fournested sums which each run over 2D grids whose x and y coordinates havebeen suppressed in the written form of the equation for simplicity (thisnotational simplicity in the equations having been followed by defaultthroughout this invention description). However, despite the nominalcomplexity of FIG. 47H, the invention is able to evaluate it with ashort fixed sequence of operations that each exhibit linear ornear-linear area scaling, thus achieving near-linear scaling overall, aswill now be discussed. Referring to FIG. 47I, the first operation inthis sequence is to carry out the summation over the constraints i,providing what will be referred to as a summed constraint coefficientmap G that is indexed by bitmap coordinates s. Although the originalsummation over s in FIGS. 47D-47H is short range (or, more particularly,sparse), reflecting the fact that the interpolations needed to obtainany one of the few intensity values that drive the ith constraint willonly involve a small number of pixels in the intensity bitmap (asdiscussed above), the s summation in FIG. 47I is no longer sparse. Thisis because the summation over i for a particular s value is preferablycarried out after investing logarithmic compute time in inverting themap between i and the particular s value, i.e. identifying the sparseset of constraints within the overall range of i whose intensityinterpolations involve pixel s. The resulting set of s values will thengenerally cover the entire bitmap after all constraints have been dealtwith, but the i summation for each particular s value only involves asmall number of constraints, whose count does not increase with fieldsize. Thus, constraint coefficient map G can be calculated for allvalues of s with near-linear area scaling. As discussed, the c_(i)constraints also include the objective C₀, and map G therefore collapsesinto a single gridded function the terms which quantitatively expressthe lithographic goals and requirements of the problem formulation.

Next, an FFT-based discrete convolution is used to calculate thequantity

that is defined in FIG. 47J as the convolution of constraint coefficientmap G with a gridding of the intensity kernel t″ of the loxicoherentsystem; and one can refer to

as an adjoint constraint map. FFTs allow this discrete convolution toattain near-linear scaling. We then calculate the product of

with convolution K; more specifically, we calculate the product of

with the real part of K, and with the imaginary part, to obtain thequantities denoted U′ and U″ that are shown in FIG. 47K. The summationsover index k in FIG. 47K then become convolutions with a gridding of themask filter kernel t′ of the loxicoherent system, and these convolutionscan be evaluated using FFTs to obtain gradient map components H′ and H″,as shown in FIG. 47L.

The summation over r in FIG. 47L amounts to an integration along an edgethat has been reduced to a summation over the edge-neighboringgridpoints within a coarsely rendered bitmap of the AL gradient (withthis gradient bitmap being the sum of H′ and H″ components). Asdiscussed above, the r summation is short-range, so that evaluation ofFIG. 47L for each single edge involves only a small number of operationswhose size is independent of mask area. Since the total number of edgevariables is proportional to mask area, FIG. 47L can thus be used tocalculate the AL gradient with near-linear scaling. It does so as theendpoint of a staged calculation which provides maps H′ and H″ of thegradient contributions on a grid of points. The individual elements ofthe gradient are then obtained by short-range integrations within thesummed maps. More specifically, the r summand in FIG. 47L usesinterpolation to approximate the integration over edge q, with theintegrated quantity being the differential change in the AL that wouldbe introduced by an “elemental” integrand consisting of a differentialexcursion of an infinitesimally short (i.e. “point-like”) edge fragmentthat might conceptually be introduced locally at some point along edgeq, with the total sum over r then representing the total contributionfrom all points along the edge, i.e. the sum represents the totaldifferential change in the AL contribution from the loxicoherent systemthat would be produced by an infinitesimal shift in position of theentire qth edge.

FIG. 47L thus provides the loxicoherent contribution to the gradient ofthe AL that is produced by all constraints which depend on imageintensity; more specifically, it provides the loxicoherent contributionfrom these intensity-dependent constraints to the gradient element thatcorresponds to the qth edge. Constraints to ensure maskmanufacturability will also make a contribution to the AL derivativewhen taken with respect to any mask edges that approachnon-manufacturability, and these gradient contributions may becalculated with near-linear scaling by using methods described in U.S.Pat. No. 8,719,735, as noted above. The AL gradient vector should thenbe extended to include the derivatives with respect to any auxiliaryvariables that may have been used in the problem formulation, e.g. toexpress infinity-norm metrics. Calculation of such derivatives isgenerally elementary. The sum in FIG. 47L has fixed range, and since thetotal number of edges scales linearly with mask area, the totalcomputational burden in evaluating FIG. 47L for all elements of thegradient vector achieves near-linear scaling. It will be clear to thoseskilled in the art that variations of this procedure can also beemployed with the other novel decomposition systems of the inventionbesides that of FIG. 20B, such as those described in FIGS. 34, 35, and41.

Successive calculations of the AL and its gradient may be used duringinner-loop steps to adjust the edge and auxiliary variables according tostandard algorithms that are designed to drive the AL to a maximum, asdescribed above. During outer loops, the Bertsekas procedure iteratesthese maximizations in conjunction with adjustments to the λ and νparameters in order to drive the solution to the true optimum of theconstrained problem formulation. Once the objective has been maximizedwithout constraint violation, and with valid multipliers λ, the e_(q)and auxiliary variables will essentially maximize the standard(non-augmented) Lagrangian, and an optimum solution will have beenattained.

Typically the total number of inner loop steps in the full optimization(summed over all outer loops) may be regarded as being roughlyindependent of mask area, with total iteration counts being, forexample, in the range of 20 to 200 depending on problem difficulty andthe accuracy sought, and with the number of outer loops being in therange of perhaps 3 to 10. Since the expected number of iterations islimited, the solution algorithm can be expected to achieve near-linearscaling overall.

Also, since each inner loop maximization is operationally analogous tominimization of a cost function, it will be clear to those skilled inthe art that near-linear scaling can also be achieved when loxicoherentsystems are incorporated into an optimization approach for designingmasks that is based on cost functions rather than constrainedformulations.

Like the Augmented Lagrangian, cost functions are typically formed as asum of terms that express different lithographic goals and requirements,and, from a mathematical point of view, minimizing an objective that isformulated as a cost involves only a trivial sign change from the caseof maximizing a function that expresses merit (i.e. benefit). Theoperations discussed in connection with FIG. 47 are essentiallyunchanged if C₀ and c_(i) are terms in a cost function instead of an AL.In general, formulations that involve unconstrained minimization of acost function and formulations that employ constraints will bothgenerally require steps in which a “merit function” is maximized duringtheir solution (or, near-equivalently, minimized as a cost function). Inthe unconstrained case, this maximization (e.g. of the sign-reversedcost function) yields the final solution, while in the constrained casesuch maximizations are carried out many times before the final solutionis obtained (e.g. during inner loops, with the merit function being anAugmented Lagrangian). This conclusion continues to hold true with morecomplicated optimization flows in which a series of optimizationproblems are solved: The solution procedure for each such problem willgenerally have as a key step the maximization of a merit function whichis the sum of terms that express lithographic goals and requirements(alternatively this key step may involve a near-equivalent costminimization), with many of these merit function terms being driven bythe image intensity. As a point of terminology, it should be noted thatthe term “merit function” has a number of different meanings in theoptimization literature. As used here in the description of thepresently preferred embodiments of this invention, “merit function” isessentially synonymous with “objective function” in the context ofunconstrained cost function formulations, while in the case ofconstrained formulations it refers to, e.g., the Augmented Lagrangian(but not the objective function C₀ of the constrained problem). In theformer case a cost function can be regarded as a merit function that isnegatively signed to express demerit. In general, the computationalbottleneck step in solving either kind of problem is that of maximizinga merit function.

With any of these approaches, the computational cost of carrying out theoptical portion of the ILT calculation will be proportional to thenumber of kernel convolutions used in the image decomposition. Becausethe loxicoherent systems of the invention allow a given accuracy targetto be reached using fewer kernels, the invention allows overall ILTruntime to be significantly reduced.

Reduced runtime makes ILT at full-chip scale more practical, whereasunder the runtime limitations of standard OCS the use of ILT is oftenrestricted to lithographically difficult areas in the layout, or tocritical circuit modules. This is particularly true where more complexILT formulations are concerned, e.g. constrained formulations in whichan Augmented Lagrangian is repeatedly re-minimized during an outersequence of loops, as opposed to one-time minimization of a costfunction.

If compute costs do permit an Augmented Lagrangian methodology to beapplied at full-chip scale, it becomes worthwhile to apply a techniquedescribed in U.S. Pat. No. 9,310,674, in which mask features areprovided with simpler shapes than are usually required for ILT, while atthe same time a high level of image quality is maintained withoutsignificant degradation from the reduced density of edge variables. Thisshape simplification leads to masks of lower cost, and to masks that canbe more tightly specified. Thus, one embodiment of the invention is aphotomask for optical lithography whose aperture shapes remaindimensionally compensating after a reduction in the number of apertureedges. These simplified aperture shapes can be partitioned into areduced number of elementary mask exposures (“shots”) when the mask isfabricated, thereby lowering mask cost.

As will be reviewed and improved upon here, this shape simplificationtechnique provides fine fragmentation where dense degrees of freedom areneeded to optimally control the binding constraints that gate theobjective (i.e. in order to optimally control lithographicallychallenging regions of the layout), while at the same time applyingcoarser fragmentation elsewhere. In general only a small fraction of theproblem constraints will turn out to be binding, but this small subsetis not known in advance, and for this reason the simplification of edgefragmentation is carried out dynamically, as will be discussed.

Since binding constraints only arise sparsely, the dynamic fragmentationtechnique will only need to apply dense fragmentation sparingly,allowing the overall fragmentation count in the full layout to besignificantly reduced, assuming that the associated optimizationmethodology can be applied over a large portion of the chip area.However, a complex constrained ILT formulation is involved (as will bediscussed), and as a result the total number of iteration adjustmentsthat are needed to converge the mask solution will typically be roughlyan order of magnitude larger than is needed with OPC, and full-chipcompute time even for OPC is already quite costly, if current OCSdecomposition is used. However, the novel loxicoherent systems of theinvention can significantly decrease the compute time needed to applyconstrained ILT formulations at full-chip scale.

Modern IC masks are written using electron-beam (“e-beam”) tools thatexpose a resist-coated mask blank. Current e-beam mask-writers arealmost always of the so-called Variable Shaped Beam (VSB) type, whichcan flash the blank with a sequence of elemental shapes, with theseelemental shapes being formed by varying the cross-sectional shape ofthe beam (“footprint”) during each flash, with the allowed beam shapestypically consisting of rectangles whose length and width areadjustable, or 45° triangles. Each flashed e-beam exposure of anelemental shape is referred to as a “shot”. Considering the case ofManhattan masks for simplicity, each feature (e.g. Manhattan polygon) inthe mask layout is partitioned into rectangles (i.e. rectangular shots)in order to write the mask. The total time needed to write the mask isstrongly correlated with the total number of rectangles in thepartitioned layout (this total being referred to as the “shot count”).Reduction of the shot count will lower the cost of the mask, due partlyto a reduction in mask-writer utilization, and partly to an easing offabrication stringency that occurs when write-time is shortened. Thequality that can be achieved in lithographic masks is partly gated byinevitable imperfections in the control of mask positioning over thefull duration of the mask writing session, and to imperfect stability inthe resist response over this time interval. Shorter write times fromreduced shot count therefore make it easier to meet specifications formask pattern positioning and sizing.

Shot count is correlated with the number of edge fragments in the masklayout, and U.S. Pat. No. 9,310,674 teaches how the number of edgevariables can be dynamically pruned in an Augmented Lagrangianformulation, without significantly compromising solution quality.Standard methods for adjusting the λ and ν parameters (discussed inconnection with FIG. 47A) during outer loops will continue tosuccessfully converge the AL to the true Lagrangian of the constrainedproblem if the variable set is adjusted after a working AL solution hasbeen maximized at the end of each outer loop cycle, as discussed in U.S.Pat. No. 9,310,674. If fragment density at the beginning of the full setof loops is initialized at the relatively high levels that areconventionally employed during ILT (since dense fragmentation maximizesthe lithographic performance benefit that ILT provides), the fragmentcount within most portions of the layout can safely be reduced betweensuccessive outer loops, but high fragment density must be maintained inmask regions that prove critical to lithographic performance. Forexample, if maximization of lithographic process window is the objectiveof the problem formulation, the most critical layout regions will bethose which first fail as process fluctuations reach the boundaries ofthe process window. U.S. Pat. No. 9,310,674 shows how a process windowmaximization goal can be quantitatively represented by a C₀ objectivefunction that is defined in terms of auxiliary variables, with theseauxiliary variables then being driven to represent the process windowattained by a lithographic image by means of a set of c_(i) constraintsthat are applied at a large number of sample points within the image. Invirtually all cases only a small fraction of these c_(i) constraintswill prove to be binding on the process window. Adjustments at othersampled image locations will only influence the binding constraints atthe level of weak long-range tails in an optical proximity response, andthese weak impacts are relatively easy to correct using even a prunedvariable set, if this pruning is carried out in accordance with theinvention.

For this reason the quality of the final solution will usually not besignificantly degraded if very short connector edges in non-criticalmask polygons are deleted from the interim working solution, with thetwo parallel edges that the deleted edge formerly connected then beingmerged into a new single edge during subsequent refinement of theworking solution. This change in the set of edge variables may becarried out after the AL has been converged to a maximum at the end ofone cycle of inner loops, i.e. at the termination of each outer loop,with the new variable set being used during the next outer loop. Thelength threshold for deletion of short edges is referred to as adeletion threshold.

Such a deletion process to reduce fragment density would becomedeleterious in layout regions that prove critical to lithographicperformance, e.g. in regions that turn out to gate the achieved processwindow. U.S. Pat. No. 9,310,674 teaches that large fragment density canbe recovered in critical areas by using a gradient map to create newfragments where needed. The gradient maps considered in U.S. Pat. No.9,310,674 are based on OCS kernels (referred to therein as SOCSkernels), but similar considerations apply with gradient maps thatcontain contributions from the loxicoherent systems of this invention(e.g. per FIG. 47L), as will now be explained. The AL will be maximizedat the conclusion of an outer loop cycle, and the integral of thegradient map along the full length of any edge will therefore be zero(assuming that the position of each edge is defined by an independentproblem variable). However, this zeroed net derivative will generallyresult from a balancing of regions of positive derivative along the edgewith regions of negative derivative. Edges containing a contiguousregion of sufficiently large (in magnitude) positive or negativederivative should preferably be split in two, with the originalcontrolling variable for the edge being replaced by new variables thatcontrol the positions of the newly created edges during the next cycleof loops. In particular, the original edge variable can be replaced by anew variable for each newly independent section of the now-split edge,and also a new variable for the newly introduced edge that connect thetwo split sections. The threshold on integrated derivative magnitudethat governs this edge creation step is referred to as an insertionthreshold.

This approach can be extended by further changing the variable set in away that aligns the corners of parallel edges which approximately faceeach other from opposite sides of mask shapes. It is known that whenmask shapes are partitioned into shots for the VSB mask-writer (e.g.when Manhattan shapes are partitioned into elemental exposedrectangles), it is efficient to capture a portion of the shape by usinga rectangle which has an edge that crosses from one corner of the shapeto a shape corner on the opposite side, so long as these two cornersshare a common coordinate value. For example, if a shape contains twovertical edges that approximately face each other across the shape, andif both of these edges have bottom endpoints that share the same ycoordinate value, it will be efficient when partitioning the shape intorectangular shots to include a rectangle whose edge spans the shapehorizontally across the locations of the two bottom endpoints. If, onthe other hand, the y coordinates of these two endpoints were insteadshifted slightly apart, an extra sliver region within the shape wouldthereby be delineated, requiring the insertion of an additional narrowrectangular shot to fully partition the shape. Shot count is thereforereduced by aligning the endpoints of edges that partially face eachother across shapes, but such alignment is undesirable at the sparse setof critical locations that bind the solution, where a large density ofadjustable degrees of freedom should be maintained.

A strategic alignment of edge endpoints where appropriate (i.e.non-critical) can be obtained by using an improved version of theabove-described methodology for dynamically adjusting edgefragmentation. More specifically, to obtain a mask exhibiting reducedshot count while providing strong lithographic performance, the requiredperformance goals and requirements can be specified in a constrainedproblem formulation that is solved using an Augmented Lagrangian method,with the set of edge variables being redefined at the commencement ofeach cycle of outer loops (optionally excluding the first outer loopcycle) in such a way as to lock into alignment the coordinates ofsuitable edge endpoints. In particular, the endpoint coordinates thatare suitable for locking may first be adjusted for exact equalization,and then kept equal by using a single common variable to control thecoordinate of both endpoints during the next set of inner loops. In apreferred embodiment the endpoints of parallel edges within each shapethat partially face each other across the shape may be brought intoalignment whenever the difference between the coordinate values of thetwo endpoints is lower than a threshold (referred to as a lockingthreshold). The common coordinate given to the newly locked endpointscan be initially set to the midpoint of their coordinate values prior tolocking, as weighted by the lengths of the corresponding two connectoredges that also intersect the endpoints of the two facing edges (e.g.,so that if the endpoint of one facing edge has a long edge connecting toit, while the edge that connects to the aligned endpoint of the otherfacing edge is short, it is appropriate to apportion a larger share ofthe equalization adjustment to the latter endpoint when bringing the twoendpoints into exact alignment). Since the endpoint coordinates arecontrolled by the positions of the two connecting edges, alignment ofthe endpoints can then be maintained by using a single common variableto control the future excursion adjustments that are made to bothconnector edges during later optimization loops, instead of assigningindependent variables to each connector.

In order that the solution provide strong lithographic performance, itis desirable that coupled endpoints be unlocked in regions that provecritical, e.g. in the vicinity of a sample point constraint that turnsout to be binding at the solution. After the AL is maximized at the endof an outer loop cycle, its derivative with respect to each problemvariable will be zero. This means that where two endpoint coordinates offacing edges have been locked to a common value, the total integratedgradient along the lengths of the two edges that connect to these lockedendpoints will be zero. However, this zero-valued total will generallyresult from a positive integral along one connector being canceled by anegative integral along the other. The endpoints should preferably beuncoupled during the next set of loops if the magnitude of theintegrated gradient along each single connector exceeds a threshold(referred to as an unlocking threshold). The thresholds used for lockingand unlocking, or for edge insertion and edge deletion, can be set byexperimentation with small layout areas, or these thresholds may bechosen in such a way as to maintain a target edge variable count duringthe loops, or to progress toward such a target. Since the deletion andlocking thresholds are preferably small, the thresholding tests forinsertion and unlocking can be applied with reasonable accuracy tocandidate pairs of connected edges or facing edges where deletion of theconnector, or locking of the endpoints, is pending but has not yet beenexecuted, i.e. to pairs that meet the criteria for deletion or locking.If these candidates also meet the criteria for insertion or unlocking,the deletion or locking step should not actually be carried out.

By using loxicoherent systems to speed the execution of each iterationloop, the above procedure can feasibly be applied over a larger portionof the full layout, leading to a significant reduction in the total shotcount within the layer, ultimately providing a mask whose shapes can bewritten in a shorter time with greater fabrication stability, whilestill providing strong dimensional compensation.

In addition to its use for overall mask design, the various ILTmethodologies described above can further be used to improve thereconciliation or ‘stitching’ of mask features at the boundaries ofregions. Another ILT application is in so-called ‘hot spot correction’,where direct optimization and enforcement of lithographic metrics isused to improve problematic areas of a first-pass mask that has beendesigned on a preliminary basis by a simpler method like OPC, forexample using subsequent ILT to improve mask shapes where thedimensional compensation achieved by the simpler method has beenfounding wanting, e.g. found to be overly sensitive to processvariability, or found to be significantly obstructed by maskmanufacturability rules. Since use of the computationally intensive ILTmethodology is reserved for areas of greatest need, a very advantageouscompromise can be made between the compute cost of mask design and thelithographic performance obtained.

It will also be clear to those skilled in the art that the invention canincrease the speed of other standard computational lithographyapplications besides OPC and ILT, such as mask design verification.

More generally, the embodiments of this invention can be expected tohave utility whenever the determination of partially coherentlithographic images is required over large areas, a task for which priorart coherent system decomposition may be considered to have only limitedsuitability. As discussed, lithographic sources do generally come closerto the coherent limit than the incoherent limit, since their directionalcontent tends to be somewhat sparse, and this does increase theconvergence accuracy of coherent decomposition. Nonetheless,lithographic sources typically contain sufficiently extended content asto require some tens of coherent systems in order to match the imagesthey produce even to the 1% level (see, e.g., FIG. 28). OCS essentiallydeals with this complex non-coherent behavior by the largely genericapproach of least-squares fitting a series of Mercer terms to the exactTCC (via eigendecomposition). As has been discussed, each loxicoherentsystem will also generally allow determination of a least-squaresoptimal fitting kernel, but loxicoherent systems go beyond this inoffering a rich variety of different structural forms which may bestrategically selected from in order to explicitly match variousdistinctive TCC content that is characteristic of a given partiallycoherent imaging system; e.g., allowing, as non-limiting examples, thechoice of the FIG. 20B form to match the fin-like residual that istypically dominant in the TCC error that remains after extraction ofcoherent systems, or the use of the FIG. 20I or 35A-35C forms to matchlow-frequency off-diagonal content in the residual TCC, or the use ofthe FIG. 34L form to match residual TCC content arising at the bandedgeof the circular pupil, or the use of the FIG. 41C form to matchcritical-axis content in the TCC that is heavily sampled by typicallithographic masks. In many of these cases the loxicoherent systems canuse one (or more) constituent kernels to closely match distinctive TCCcontent that is recalcitrant to matching by OCS (or can even exactlymatch this content), while simultaneously providing a least-squaresoptimal minimization of residual TCC error over the full Hopkins domainusing another constituent kernel, thereby significantly easing acomputational bottleneck in accurately providing dimensionallycompensated mask shapes.

The embodiments of this invention thus provide in one aspect thereof atool configured to input integrated circuit (IC) circuit patterns so asto form one or more IC fabrication masks, where the tool includes andencompasses a method and structure and computer program for implementinga decomposition-based analysis of data representing a mask. The toolproduces an output database or output data stream in which thedimensions of the mask shapes are compensated on the basis of the imagecontent in the vicinity of each shape when the mask is projected duringoptical lithography. The tool superimposes a sum of images from a set ofcoherent systems and a sum of images from a set comprised of at leastone loxicoherent system. Each loxicoherent system is a compound systemcomprising a paired coherent system and incoherent system that act insequence, with the output of the constituent coherent system being inputas a self-luminous quantity to the constituent incoherent system, andwith the output of the incoherent system then serving as the output ofthe loxicoherent system.

It is again noted that the novel loxicoherent decomposition systems thatare a feature of this invention are not coherent systems. Thedecomposition systems of the invention produce intensities which arelinearly summed to match the partially coherent image intensity ofinterest, where each loxicoherent system presents a richer structurethan does a prior art coherent/Mercer system. The loxicoherent systemsare in essence compound systems, giving rise to a compounded behaviorthat is fundamentally nonlinear. In most embodiments the constituentsystems of the loxicoherent system operate in sequence, with the outputof a constituent coherent system, or the summed output from a pluralityof constituent coherent systems, being passed as an internal input to aconstituent incoherent system, whose output serves as the imagecontribution of the loxicoherent system as a whole. The nonlinear effectof this sequential operation cannot generally be matched by a coherentsystem, and, for that matter, it is impossible for any coherent systemto even match the behavior of the constituent incoherent system alone,except in “pathological cases”. Thus, the systems used in this inventionwould not be classified as being coherent systems per se. However, it ispointed out that the term “loxicoherent system” also covers theDC-monolinear embodiment. In this embodiment the constituent kernelsproduce amplitudes which interfere with one another to produce theoutput intensity of the DC-monolinear system. This interference processis represented computationally by a multiplication (and not a sum), andis therefore fundamentally nonlinear. Thus, the combined behavior of thetwo constituent DC-monolinear kernels is inherently quite different fromthe prior art behavior exhibited by e.g. two coherent systems whoseoutput is summed per the OCS procedure. However, the order in which thetwo constituent kernels of the DC-monolinear system are applied does notaffect the output, whereas most of the other novel systems employed bythe invention require that their constituent systems by applied in theproper sequence.

There is also one aspect of DC-monolinear systems to which the term“coherent” might be applied in a certain sense, but the coherentbehavior involved is distinctly different from that of coherent OCSsystems. A DC-monolinear system will exhibit a coherent aspect (in onesense of the term) in cases where its constituent spatial domain kernelp(x) can be considered roughly constant. In preferred embodiments thefrequency domain kernel P(f) will tend to have a narrow peak at theorigin which will bear qualitative resemblance to a delta-function, andto the extent that p(x) can then be considered roughly constant, onemight reasonably regard the computational output of the fullDC-monolinear system as being somewhat analogous to a near-linearcalculation of a coherent amplitude, as previously discussed.Nonetheless, such behavior would be quite different from that of thecoherent systems in prior art OCS, because OCS systems produce theiroutput intensity as the square of a coherent amplitude, i.e.computationally the OCS systems are represented in their intensityoutput by a quadratic (and thus nonlinear) function of the amplitude,even though the dependence is linear before being squared. Moreover, itis also true, of course, that both kernels used by a DC-monolinearsystem will be quite different from the kernels of any OCS systemappearing in a Mercer series decomposition of the TCC. Further, inpreferred embodiments the p(x) kernels of DC-monolinear systems arechosen to minimize the RMS error in matching TCC^((r)) over the fulldoubled domain (e.g. by using FIG. 41E), and in such cases p(x) willtypically deviate fairly substantially from a constant amplitude.

In general, the novel decomposition systems used by all embodiments ofthis invention are clearly distinguishable from conventionalsystems/approaches at least in view of the fact that all such discloseddecomposition systems use more than one distinct kernel function,reflecting the fact that these novel decomposition systems are compoundsystems whose output combines the outputs from a plurality ofconstituent systems in a nonlinear fashion.

FIG. 18 shows an exemplary embodiment of the present invention. A systemincludes a tool that facilitates fabrication of masks for opticallithography, where the tool can be embodied at least in part as acomputer system 700 having one or more processors 705, one or morememories 710, and one or more network interfaces 720, interconnectedusing one or more buses 730. The one or more processors 705 canimplement the processors #1-#F in FIG. 30A that operate in parallel toexecute the Group 1300 operations shown in FIG. 30C. The one or morememories 710 include a computer program 715 defined to cause thecomputer system to perform one or more of the operations describedherein. An input to the computer system 700 includes a starting mask 785(also shown in FIG. 19) which may be represented as a set of desiredsemiconductor device shapes. In one embodiment the mask information 740(which can be referred to as well as a ‘final mask 795’ as in FIG. 19)obtained by execution of the computer program 715 is output by the toolto a mask making machine 735 via link 745. The mask making machine 735makes a physical mask 750 from the mask information 740. The mask makingmachine 735 can, in some non-limiting embodiments, be an e-beam maskwriter of the Variable Shaped Beam (VSB) type that was discussed above.The photomask 750 is provided to and used by a lithography andprocessing system 760 to create device and other shapes on asemiconductor 770, such as a semiconductor wafer or substrate. Thecomputer program 715 thus contains instructions to implement the methodaccording to the present invention as shown in, for example, FIG. 30.Data representing a mask of interest that is created by the use of thetool can be stored in the memory 710 or in some other memory, and themask data created by the tool in accordance with the embodiments of thisinvention can then be subsequently read-out and processed during an ICfabrication operation. This data that can be stored and read out asneeded can be considered to represent, for example, a data assemblage ora data structure or structures that is stored on some non-transitory andcomputer-readable storage medium.

FIG. 19 is a diagram of an exemplary non-limiting embodiment of the maskmaking tool in accordance with this invention that in this casecomprises an apparatus that includes an OCS system engine 780 thatreceives the starting mask 785 which is typically the set of desiredshapes for the printed semiconductor devices. The starting mask 785 canbe organized into separated regions of mask content, and the OCS engine780 can include a frame generation function/module 787 configured topartition each region into overlapped frames of mask data (see, forexample, blocks 1102A, 1102B and 1102C of FIG. 30A). The OCS engine 780outputs a full TCC to an input of a loxicoherent system engine 790 thatis constructed and operated in accordance with embodiments of thisinvention to provide a final mask 795. In this embodiment the OCS engine780 and the loxicoherent system engine 790 operate in sequence with theOCS engine operating first. The engines 780 and 790 can be constructedfrom hardware that is configured so as to execute the operationsdescribed above and that are shown generally in FIG. 30. For example,the OCS system engine 780 can be configured with specialized circuitrythat executes at least the blocks 1202-1214 shown in FIG. 30B, and theloxicoherent system engine 790 can be configured with specializedcircuitry that executes at least the blocks of the step Group 1300 shownin FIG. 30C. For example, the loxicoherent system engine 790 can containan array of the parallel connected frame processors as in FIG. 30A andpossibly also other circuitry (e.g., dedicated logic elements and statemachines) configured to perform, e.g., as in FIG. 30C: determiningloxicoherent system contributions to the image intensity at target edgepositions by applying the intensity kernels to squared masktransmissions that have been filtered by the mask filters; determiningthe image intensity at target edge positions by adding the loxicoherentcontributions to the sum of intensities from the preferred coherentsystems; moving mask fragments adjacent to target edge positions whoseintensity is lower than the intensity at the edge of the anchoringfeature in a direction towards the ‘darker’ side of the adjacent targetedge; moving mask fragments adjacent to target edge positions whoseintensity is higher than the intensity at the edge of the anchoringfeature in a direction towards the ‘brighter’ side of the adjacenttarget edge; transferring to others of the parallel connected processors(those handling adjacent frames that are overlapped by this guard band)the iterated positions of fragments within the guard band of the framebeing processed, and using position data from the guard bands of otherframes that have similarly been transferred from the adjacent-frameprocessors to unify and harmonize the positions of fragments in theexterior guard band of the frame being processed before commencing thenext iteration cycle; and terminating the adjustment cycles when theintensities at all target edge positions match that of the anchoringfeature to within a tolerance.

In some embodiments the data processing system or systems and CPU(s) andmemory and storage device(s) can be instantiated in whole or in part asone or more virtual computing systems in a cloud computing environment.

In a further embodiment of the present invention a method, for exampleas in FIG. 30, may be provided as a service to a mask designer forobtaining, characterizing, and verifying a mask design.

In general any combination of one or more computer readable medium(s)may be utilized. The computer readable medium may be a computer readablesignal medium or a computer readable storage medium. A computer readablestorage medium may be, for example, but not limited to, an electronic,magnetic, optical, electromagnetic, infrared, or semiconductor system,apparatus, or device, or any suitable combination of the foregoing. Morespecific examples (a non-exhaustive list) of the computer readablestorage medium would include the following: an electrical connectionhaving one or more wires, a portable computer diskette, a hard disk, arandom access memory (RAM), a read-only memory (ROM), an erasableprogrammable read-only memory (EPROM or Flash memory), an optical fiber,a portable compact disc read-only memory (CD-ROM), an optical storagedevice, a magnetic storage device, or any suitable combination of theforegoing. In the context of this disclosure a computer readable storagemedium may be any tangible medium that can contain, or store a programfor use by or in connection with an instruction execution system,apparatus, or device.

A computer readable storage medium, as used herein, is not to beconstrued as being transitory signals per se, such as radio waves orother freely propagating electromagnetic waves, electromagnetic wavespropagating through a waveguide or other transmission media (e.g., lightpulses passing through a fiber-optic cable), or electrical signalstransmitted through a wire.

Program code embodied on a computer readable medium may be transmittedusing any appropriate medium, including but not limited to wireless,wireline, optical fiber cable, RF, etc., or any suitable combination ofthe foregoing.

Computer program code for carrying out operations for aspects of thepresent invention may be written in any combination of one or moreprogramming languages, including an object oriented programming languagesuch as C++ or the like and conventional procedural programminglanguages, such as the “C” programming language or similar programminglanguages. The program code may execute entirely on a single localcomputer, partly on the local computer, as a stand-alone softwarepackage, partly on the local computer and partly on a remote computer orentirely on the remote computer or server. In the latter scenario, theremote computer may be connected to the local computer through any typeof network, including a LAN or a WAN, or the connection may be made toan external computer (for example, through the Internet using anInternet Service Provider).

Aspects of the mask design and configuration tool that provides a maskhaving dimensionally compensated shapes that is a feature of the presentinvention are described with reference to flowchart illustrations and/orblock diagrams of methods, apparatus (systems) and computer programproducts according to embodiments of the invention. It will beunderstood that each block of the flowchart illustrations and/or blockdiagrams, and combinations of blocks in the flowchart illustrationsand/or block diagrams, can be implemented by computer programinstructions. These computer program instructions may be provided to aprocessor of a general purpose computer, special purpose computer, orother programmable data processing apparatus to produce a machine, suchthat the instructions, which execute via the processor of the computeror other programmable data processing apparatus, create means forimplementing the functions/acts specified in the flowchart and/or blockdiagram block or blocks.

Computer readable program instructions described herein can bedownloaded to respective computing/processing devices from a computerreadable storage medium or to an external computer or external storagedevice via a network, for example, the Internet, a local area network, awide area network and/or a wireless network. The network may comprisecopper transmission cables, optical transmission fibers, wirelesstransmission, routers, firewalls, switches, gateway computers and/oredge servers. A network adapter card or network interface in eachcomputing/processing device receives computer readable programinstructions from the network and forwards the computer readable programinstructions for storage in a computer readable storage medium withinthe respective computing/processing device.

These computer program instructions may also be stored in a computerreadable medium that can direct a computer, other programmable dataprocessing apparatus, or other devices to function in a particularmanner, such that the instructions stored in the computer readablemedium produce an article of manufacture including instructions whichimplement the function/act specified in the flowchart and/or blockdiagram block or blocks.

As such, it should be realized that a computer readable medium canpresent a tangible carrier for information that is recorded or otherwiseimpressed on or in the computer readable medium, where the informationis configured to cause a programmable device to implement the tool thatincludes in part the loxicoherent system methods, apparatus and routinesof this invention. The resulting combination of the tangible,non-transitory computer readable medium and the information storedtherein or thereon is clearly, in at least one aspect thereof, anarticle of manufacture. The article of manufacture, which can be acomponent part of the tool in accordance with this invention as depictedin FIGS. 18 and 19, is usefully configured to aid in converting andtransforming a first object, i.e., the initial or starting mask 785,which may be represented as the set of desired semiconductor deviceshapes, into a second object, i.e., the final mask 795 that can be usedduring the fabrication of semiconductor circuits and structures.

The computer program instructions may also be loaded onto a computer,other programmable data processing apparatus, or other devices to causea series of operational steps to be performed on the computer, otherprogrammable apparatus or other devices to produce a computerimplemented process such that the instructions which execute on thecomputer or other programmable apparatus provide processes forimplementing the functions/acts specified in the flowchart and/or blockdiagram block or blocks.

The flowchart and block diagrams in the Figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams may represent a module, segment, or portionof code, which comprises one or more executable instructions forimplementing the specified logical function(s). It should also be notedthat, in some alternative implementations, the functions noted in theblock may occur out of the order noted in the figures. For example, twoblocks shown in succession may, in fact, be executed substantiallyconcurrently, or the blocks may sometimes be executed in the reverseorder, depending upon the functionality involved. It will also be notedthat each block of the block diagrams and/or flowchart illustration, andcombinations of blocks in the block diagrams and/or flowchartillustration, can be implemented by special purpose hardware-basedsystems that perform the specified functions or acts, or combinations ofspecial purpose hardware and computer instructions.

The terminology used herein is for the purpose of describing particularembodiments only and is not intended to be limiting of the invention. Asused herein, the singular forms “a”, “an” and “the” are intended toinclude the plural forms as well, unless the context clearly indicatesotherwise. It will be further understood that the terms “comprises”and/or “comprising,” when used in this specification, specify thepresence of stated features, integers, steps, operations, elements,and/or components, but do not preclude the presence or addition of oneor more other features, integers, steps, operations, elements,components, and/or groups thereof.

The corresponding structures, materials, acts, and equivalents of allmeans or step plus function elements in the claims below are intended toinclude any structure, material, or act for performing the function incombination with other claimed elements as specifically claimed. Thedescription of the present invention has been presented for purposes ofillustration and description, but is not intended to be exhaustive orlimited to the invention in the form disclosed. Many modifications andvariations will be apparent to those of ordinary skill in the artwithout departing from the scope and spirit of the invention. Theembodiment was chosen and described in order to best explain theprinciples of the invention and the practical application, and to enableothers of ordinary skill in the art to understand the invention forvarious embodiments with various modifications as are suited to theparticular use contemplated.

As such, various modifications and adaptations may become apparent tothose skilled in the relevant arts in view of the foregoing description,when read in conjunction with the accompanying drawings and the appendedclaims. As but some examples, the use of other similar or equivalentmathematical expressions may be used by those skilled in the art.However, all such and similar modifications of the teachings of thisinvention will still fall within the scope of this invention.

What is claimed is:
 1. A computer-controlled tool configured to processinput data representing integrated circuit patterns of a semiconductorfabrication mask to be used in projection lithography, comprising: atleast one data processor configured to apply a dimensional compensationto circuit pattern shapes based on an intensity pattern produced in aprojected lithographic image, where the intensity pattern is determinedby performing an optimal coherent systems process on the input datausing coherent optimal coherent systems kernels derived from at leastone Hopkins bilinear transmission cross coefficient; and performing adecomposition process on at least a portion of the input data using atleast one compound loxicoherent system in which a constituent coherentsystem is paired with a constituent incoherent system to form the atleast one compound loxicoherent system, and where at least one kerneldecomposition is made along an axis that is slanted between two domainsof a Hopkins bilinear model to determine an aperture of the incoherentsystem.
 2. The computer-controlled tool as in claim 1, where an inputfor both the constituent coherent system and the constituent incoherentsystem is a frame of integrated circuit shapes constituting a portion ofa mask region from the input data.
 3. The computer-controlled tool as inclaim 1, where, in the at least one compound loxicoherent system, theconstituent coherent system and the constituent incoherent system act insequence, where an output of the constituent coherent system is input asa self-luminous quantity to the constituent incoherent system with whichthe constituent system is paired, and where an output of the constituentincoherent system is an output of the at least one compound loxicoherentsystem.
 4. The computer-controlled tool as in claim 1, where lensapertures of coherent systems used in the optimal coherent systemsprocess are Fourier transforms of optimal coherent systems kernelsobtained by carrying out an eigendecomposition process on a fulltransmission cross coefficient.
 5. The computer-controlled tool as inclaim 4, where respective lens apertures of the constituent coherent andincoherent systems in each compound system of the at least one compoundloxicoherent system are obtained by isolating a residual transmissioncross coefficient that remains after a chosen set of coherent kernels,of the coherent systems used in the optimal coherent systems process,are extracted from the at least one Hopkins bilinear transmission crosscoefficient; and performing at least one decomposition process on theresidual transmission cross coefficient using the at least one compoundloxicoherent system.
 6. The computer-controlled tool as in claim 4,where a first loxicoherent system of the at least one compoundloxicoherent system is selected to match portions of the at least oneHopkins bilinear transmission cross coefficient that are recalcitrant tomatching by a paired optimal coherent systems component.
 7. Acomputer-implemented method to process data representing a semiconductorfabrication mask, comprising: performing an optimal coherent systemsprocess on the data using optimal coherent systems kernels derived fromat least one Hopkins bilinear model; and performing a decompositionprocess on at least a portion of the data using at least oneloxicoherent kernel, in which at least one kernel decomposition is madealong an axis that is slanted between two domains of the Hopkinsbilinear model.
 8. The method as in claim 7, where an input for the atleast one loxicoherent kernel is a frame of integrated circuit shapesconstituting a portion of a mask region from the data.
 9. The method asin claim 7, where lens apertures of coherent systems used in the optimalcoherent systems process are Fourier transforms of optimal coherentsystems kernels obtained by carrying out an eigendecomposition processon a full transmission cross coefficient.
 10. The method as in claim 9,where at least one lens aperture of the at least one loxicoherent kernelis obtained by isolating a residual transmission cross coefficient thatremains after a chosen set of coherent kernels, of the coherent systemsused in the optimal coherent systems process, are extracted from the atleast one Hopkins bilinear model; and performing at least onedecomposition process on the residual transmission cross coefficientusing the at least one loxicoherent kernel.
 11. The method as in claim9, where a first loxicoherent kernel of the at least one loxicoherentkernel is selected to match portions of the at least one Hopkinsbilinear model that are recalcitrant to matching by a paired optimalcoherent systems component.
 12. A data assemblage stored on or in anon-transitory computer-readable storage medium, the data assemblagerepresenting mask data for use in fabricating an integrated circuit, thedata assemblage being created by a process that comprises performing anoptimal coherent systems process on the mask data using optimal coherentsystems kernels derived from at least one Hopkins bilinear model; andperforming a decomposition process on at least a portion of the maskdata using at least one loxicoherent kernel, in which at least onekernel decomposition is made along an axis that is slanted between twodomains of the Hopkins bilinear model.
 13. The data assemblage as inclaim 12, where an input for the at least one loxicoherent kernel is aframe of integrated circuit shapes constituting a portion of a maskregion from the mask data.
 14. The data assemblage as in claim 12, wherelens apertures of coherent systems used in the optimal coherent systemsprocess are Fourier transforms of optimal coherent systems kernelsobtained by carrying out an eigendecomposition process on a fulltransmission cross coefficient.
 15. The data assemblage as in claim 14,where at least one lens aperture of the at least one loxicoherent kernelis obtained by isolating a residual transmission cross coefficient thatremains after a chosen set of coherent kernels, of the coherent systemsused in the optimal coherent systems process, are extracted from the atleast one Hopkins bilinear model; and performing at least onedecomposition process on the residual transmission cross coefficientusing the at least one loxicoherent kernel.
 16. The data assemblage asin claim 14, where a first loxicoherent kernel of the at least oneloxicoherent kernel is selected to match portions of the at least oneHopkins bilinear model that are recalcitrant to matching by a pairedoptimal coherent systems component.